They spend that time being observed by the sisters who are trying to decide if they want them or not. It takes a while to decide if taking a weakling like Elin or Daigian is offset by her mental abilities. Note that those two went for the White & Brown, where their One Power skills are least relevant to their tasks. For the other Ajah functions, no matter how little they have to do with the Power, strength & the authority it imparts to a sister give the impression of capability. Grays, for example, probably have trouble taking a weakling seriously as a negotiator or mediator; Blues would doubt the ability of a weakling to make any headway at her causes and the other three could not do their jobs without the Power, and can't see how much you can Heal or how powerful a channeler you can counter if you can barely handle enough of the Power to reach the shawl. I would assume that among the Whites and Browns, they make exceptions for the hierarchy, if they have ANY intellectual integrity, to prevent status issues from distorting their dicussions and scholastic works, with stronger sisters squelching the superior mental work of their lessers. Hence they would be the most likely to finally break down and accept a long-time initiate. Also, note that spending two decades in the Tower is enough time for a Accepted with the freedom to choose her own studies (how many weaves can there be and does it REALLY take 20 years to learn mere channeling? I can't believe that is the hold-up) to build a scholastic reputation which would induce those Ajah to accept her.

I don't think that is at all typical of the time. Elin Warrell has taken that long, but she is not presented as average or normal, rather she is offered as an extreme case. A supporting detail, IMO, is the number of women who were apparently Siuan's & Leane's peers as trainees. They wouldn't have so many stories of mischief and times together unless their progression was not far behind to Moiraine's & Siuan's. Otherwise, Siuan would only have known them for a relatively brief time in their training, when she showed up, passed them by, and moved on. For someone whipping through their training at 3x the normal speed, making friends with people who started so far ahead of you and fell so far behind so fast would be difficult at best, and highly unlikely to be the source of so many shared exploits and common knowledge of secret pastimes. If Siuan's pace of advancement was so abberrantly faster than theirs, their friendships would have been founded more on shared time as full sisters.

But why would being at their full potential have anything to do with it? As long as they have mustered sufficient strength to channel to whatever standard the Tower requires, they should be accepted for the shawl if their level strength is an issue (which point you seem to be making in the original post). I don't think it is the strength hierarchy, or not directly. Rather, I think the mindset behind using strength as a criterion for the hierarchy is what influences them.

My perception of the Tower advancement process is that it is much less strictly formalized and standardized, and more of a process of personal development. We can't think of it in terms of contemporary schooling, which might be comparable to an education by assembly line, but rather as a holistic process that might extend the analogy to education by individual mastercraft. That's why the seemingly absurd tests to advance are of the kind they are. They aren't REALLY exams to qualify for the next step, rather they are in the way of a final initiation step, that the sisters do not let anyone take until or unless they have decided that they are okay with the woman in question advancing. THAT, IMO, would be a far better explanation for the rapid advance of notably strong women like Elaida, et al. Just about any sister COULD advance that fast, but they really take note of the strong ones and pay them more attention, jump on infractions with an eye for correcting them faster, and give them lessons and advice on sisterly conduct more freely and willingly. They are quicker to spot signs of readiness for advancement and correct flaws (and note how well they are taken to heart) in the more promising candidates, and are probably more willing to write off pranks and misdeeds as youthful exuberance, rather than signs of crippling immaturity. When Siuan gets caught fishing off the Tower grounds without permission, they rationalize it as "no harm, no foul" and make an effort to pound such behavior out of her. When Elin is caught sneaking immature books out of the library, they sigh in dismay, wonder when she will ever learn, and forget about her until she next pops up on their radar (probably for another infraction). When initiates are reviewed to see who's ready to move up and who isn't, the powers that be on such matters will be looking for specific reasons to advance the initiates (Aes Sedai stature being a privileged and elite position that enhances their self-importance to withhold), as opposed to letting them all advance by default after ensuring there is no good reason why they should be held back as is common in real world training and educational programs (who are not admitting prospects into an exclusive fold, but rather getting the kids out of their hair as soon as they can be sure they won't be embarrassed by their protegees' ignorance, or else training their future colleagues and thus will benefit from getting them into the workforce ASAP). With that mindset in place, the stronger initiates benefit from greater attention. When the sisters wonder whom should be raised next, it is the stronger girls, who are of more interest and the beneficiaries of more attention and scrutiny who spring to mind, while the middling and weaker girls get overlooked for years.

Another reason, and more important, I fell, is that a large part of what determines readiness to move on is dedication to the Tower and Aes Sedai as a whole, as well as maintenance of that stregth and unity of the Tower. For the stronger women who will hold high places in the Tower and who will benefit the most by the hierarchy, that is less of an issue in their assessment. But for the middling ones, or the just-strong-enough sisters who will have to spend 300 years serving tea after spending years training and working towards the goal of someday being the equal of all of her teachers and a powerful woman of great authority, only to be handed the shawl and made to understand that would never happen? How long before such women decide to break off and go their own way, or start agitating for changes in the status quo? The Tower will want to be absolutely certain of their loyalty, dedication, and most importantly, subordination to the rules, practices and traditions of the White Tower. They are not going to worry so much about the stronger sisters, because at the level Aes Sedai operate, the stakes are always very high and their work takes on great importance to a sister. If you have an agenda or task that you believe is extremely important, why would you piss away a system that gives you authority over a great many sisters, over some abstract concept of fairness? Even if they were so inclined to question the Tower, for example, Moiraine and Siuan had the job of finding the Dragon Reborn dropped in their laps. Why, with a task upon which the fate of the world could hinge, would they stop and say "Hey, you people with five times our lifespan as sisters should not be listening to us 22-year olds just because of a random accident of birth! Let's change this system." It's all very well for Nynaeve to suggest as much to Daigian at a moment when the latter is demonstrating a more or less useless skill to her and is an object of pity, but ask Nynaeve her opinion of the system the next time the arrogant Daigian attempted to blow Nynaeve off or scoff at her for being a wilder or doubt her legitimacy as a sister, when Nynaeve is trying to help the Dragon Reborn or foil another Forsaken plot or get through a fight without getting an Asha'man killed because the leader of a link can't be bothered to hold as much of the Power in readiness for an attack as possible.

Yes, but assuming that motivation does not make sense. As I say, better to bring them into the hierarchy when they are still low and let them ascend as their strength grows, even (or better yet, especially) if it takes as long as a decade or two to reach that point. If that IS their motivation, they are going about it in a screwy way.

But it does not seem to have worked out that way.

And she'll grow into that notion as she progresses through the hierarchy. Since strength never decreases, no one moves downward, and thus the hierarchy never "punishes," so in general, a positive perception of the hierarchy is maintained.

Why? The Aes Sedai are united against the world, but internally, have factions and compete to get influence and support for their agendas. Why surrender a tool like the hierarchy which gives stronger sisters advantages? Rather than undo an order that places them on top and enables their causes and agendas, it would occur to a strong sister to instead patronize those wise and capable sisters and make sure their views receive due respect. Such a tactic would allow the strong and sensible sisters to have their cake and eat it too. Then they can justify leaving the system in place "because I am using that authority for GOOD causes! I only squelch the idiots who say stupid things or push foolish agendas. I am fair and considerate in the use of my authority and without me up here at the top, who is to say anyone would listen to my friend the scholar? She IS kind of absent-minded to be working on maintaining her influence and keeping her name in the public eye."
As it is, the weaker sisters join with the stronger in temporary factions and alliances to get agendas through and the stronger look out for the weaker or pass them information, or help them lay plans. That is what Delana meant when she was talking with Siuan in the beginning of the PoV where she meets Aran'gar. It was not physical dangers, but rather she was offering her friend support against women with greater strength. No doubt she sincerely wanted to do right by her friend, but I would not be surprised to find that she was also looking forward to being able to pick the brain of a notorious schemer like Siuan to advance her own agendas, or get access to timely information from the Tower-in-exile's unofficial spymistress.
We see that the experienced Sitters in the Hall in Salidar gravitate into factions around influential leaders, which suggests this is the normal practice, and it is only the newbies who don't know how things work that go their own way. Cadsuane appears to have a relationship like I described with Daigian and Kumira, making use of their minds and at the same time protecting them from being bullied, if only by bullying their betters so, that they don't have the time or gumption to pick on Daigian. Or maybe even just for the schadenfreude of hanging around Cadsuane and getting to see her treat the middling to high sisters the way those same women routinely treat the weaklings.

THAT, I think, is what is behind the exaggerated duration of the training. They are ensuring that the newly raised sisters will not question things too much, or be too dissatisfied with their new status. Get them used to thinking of Tower rules and Tradition as the arbiter of all, and then peer pressure and the inexperience and uncertainty of new sisters, raised up suddenly to such a dizzying height, is enough to keep them in line until they get into the habit of properly behaving as a sister ought (as Pevara and Seaine suggest is the normal MO in their first conversation). As I noted above, the strongest girls have too much to gain from the systemt to throw it over, so they can be advanced at a more reasonable speed, and inculcated in the traditions and customs of full sisterhood when they are young enough that it sinks in better, and if they reach the shawl when there are more sisters ahead of them than would be the case 5 years later, so much the better that there are more women to whip them into shape or hold them down until they take on the habits of normal sisters. Siuan's notably rebellious nature might also have been a factor in the Ajah finding her a strictly supervised job right off the bat. Despite her intentions to travel and see the world and do her own thing when an initiate, the Tower is one jump ahead of Siuan and she finds herself getting jammed into the mold of the typical Tower-oriented Aes Sedai, rather than allowed to go off on her own and develop into the sort of unorthodox sister such as Cadsuane who causes so much trouble for the guardians of the status quo. Meanwhile, a middling-strong woman with a highly intelligent mind who learns really fast, and matches some of her stronger fellow Accepted or novices in skill with the Power... why, she's just the type you have to watch out for when you are trying to protect the system! Capable, but with cause to be disaffected, and especially if she shows any sign of independant thought. No, she gets held back until they are sure she won't make trouble or be disloyal once she has the shawl and becomes harder to rein in. Thus, she spends more of her formative years as a lowly novice or Accepted, and doesn't come into her full rank until she is older, more set in her habits of obediance to superior sisters, Tower law, customs and tradition and those women placed above her by the Tower. A weak sister who becomes Aes Sedai at 20-25 might reach the shawl young enough to quickly become adapt to new circumstances and shrug off authority that has only custom behind it, and learn to start ignoring women she has been thinking of as being all equal up until now. A strong sister the same age can be trusted, however, because her own self-interest will encourage her to accept the strength hierarchy in her formative years. The weaker sisters come into the shawl when they are older and better subordinated to the practice of deference, and they take longer to start seeing other sisters as peers, and by the time they are ready to stand up for themselves, they have been broken to the harness. The stronger ones, however, who get inculcated into the hierarchy younger and having the most ability to use that hierarchy to mess with the structure of the Aes Sedai, are rather made to accept the strictures of their Ajahs and the customs which restrain sisters, but which Accepted cannot be allowed to know het. The stronger sisters thus take in Aes Sedai culture younger, while the weaker are given longer conditioning in the habits of deference, and when they reach the shawl, are kept in their place by the stronger ones.