My problem is with the fact that the Aes Sedai think holding Sea Folk sisters in captivity is more important than retaining their secrets, that they will be outraged at the idea their "sisters" will be given a choice for the first time in their adult lives.

As for Nynaeve and Elayne thinking they gave away too much, sure, they only performed at the level of a normal Aes Sedai, which is a huge failure by their standards. Egwene, on the other hand, has no business holding them so strictly to account. On her own, she is not worthy to criticize their accomplishments, and as the Amyrlin, her reprimands should be by policy only, rather than personal disappointment. The policy, as you pointed out, is that the matter of knowledge and teachers is not significant enough to make a big deal over. The thing that WILL be a problem for the Tower is a wrong-headed, and even evil policy that the Tower owns everyone who has ever been written into the novice book, and they may not go home and live as they wish.

That's exactly the point! She reams them out over political issues. As for "allowing them to take the fall so her position would be safe" in the first place, you are treating Egwene's refraining from a particular wrongdoing as an active good deed, when refraining from wrongdoing is the minimal expectation for everyone. Political power and position exist for the benefit of its constituency, not the politician. Egwene refraining from doing practical harm for political expediency is not choosing one good over another, she is morally equal to the politician who draws the line at embezzlement and misappropriation of public funds. That's all well and good, but that is the expected minimum, not a noble or heroic sacrifice! And in the second place, that's not even an option for Egwene. As Thom noted in LoC, Elayne & Nyaneve going down would take Egwene down with them. Throwing them under the bus is not in it for Egwene, because their shortcomings would reflect badly on her to a Hall that does not recognize their legitimacy. As Egwene herself admits to them the first day, she raised them for political advantages to herself. Their ability to help the world and the Light was purely a coincidental benefit next to her political agenda.

So Egwene is not even refraining from doing wrong by her friends, she is refraining from committing suicide to spite her friends. Whoop de do.

What knowledge was that? By Bair's own admission, they hardly taught Nynaeve & Elayne anything, and they were more interested in reining the pair in and preventing them from doing anything. Most of what Nynaeve & Elayne learned, aside from need-searching, they did practicing on their own on the trip to Salidar. It was not the Wise Ones' teaching that enabled Nynaeve to capture Moghedian, it was a skill she and Elayne honed on their own, inspired largely by Nynaeve's fear of Egwene's assault. The Wise Ones tried to make Nynaeve promise not to attempt anything on her own in TAR, which would have precluded finding the Bowl of the Winds, among other things. Meanwhile Egwene mocked Nynaeve's refusal to give the same false promise she herself had made, and then pretended outrage at the white lie she told Melaine to keep the Wise One from trying harder to curtail her individual explorations.

But they were things she accomplished in TAR. I'd rather learn from the woman who has actually done something, than the one with an admirable report card, and no achievements to speak of beyond tormenting a friend.
But Nynaeve proved better able to master the first in spite of various advantages Egwene had in that area; Egwene sucks at the second, and that ability made exactly zero difference in the grand scheme of things, and the communication stuff only served to remind the readers that these people were all in the same story. They communicated just fine without any lessons from the Wise Ones, as they established a practice of meeting in TAR before Egwene went to the Waste. Also, that vaunted communications technique is responsible for the bargain Egwene hates so much. They went to the Sea Folk because Egwene told them they could not come back to the rebel camp yet. Since they didn't know when they'd have enough sisters to use the Bowl, they decided to lay the groundwork for its eventual use by recruiting the Windfinders. While that turned out for the best, as with any positive result from Egwene's actions, it was completely unintended by her.
No, she refused their restrictions, because whatever the benefits to her personally, it would interfere with the work she had to do, and her fight against the Shadow. The Wise Ones did not offer to teach her, they demanded a promise from her that she would refrain from spying on the Forsaken or Black Ajah, holding their aid hostage to her compliance. Nynaeve refused to give it, because she would not lie, since she intended to go about her business anyway. And the Wise Ones ought to be ashamed of themselves, refusing aid to people actively fighting the Shadow. At least when Moiraine did so, such as at Fal Dara, she had the excuse of trying to fight the Shadow elsewhere. The Wise Ones were just preoccupied with trying to contain the fallout of their allowing their people to be ignorant racists for three millenia. That's all well and good, but if that's your priority, don't interfere with people fighting the Shadow so you can feel good about your ethics. Even Siuan Sanche was willing to risk allowing untrained girls to be harmed in order to fight the Dark One's servants.
Moghedian characterized that trap as a mere trifle, that she only set because Egwene was associated with the woman she feared most, and felt only mild disappointment and curiosity at her escape. Nynaeve caught Moghedian by using the properties of TAR. When I referenced Nynaeve capturing Moghedian, why would you think I was referring to their prior encounter? As for Birgitte's intervention, she was willing to intervene, because Nynaeve impressed a Hero of the Horn, because Birgitte, unlike certain Wise One BFF of Egwene, was not willing to sit back and let someone fight the Shadow without helping. Anyway, Nynaeve learned from her first failure, and caught Moghedian spying on her, in spite of Siuan's petty bullshit and distractions, got Siuan safely away, and then when Moghedian tried to capture her and Birgitte, turned the tables using a technique she and Elayne taught themselves, after the Wise Ones refused their help because Nynaeve wouldn't put her personal safety & tired old institutional ethics ahead of the fight.