…and finds excuses for Gareth Bryne, without whom, Siuan and Gawyn would never have effected her removal and caused her to miss her chance to seize power in the confusion. See, no one follows Siuan or Gawyn. They don’t bring any armies or international reputation to her cause, so even though Bryne’s disobedience was the biggest factor in obtaining the result Egwene dislikes, she’s going to let him slide and even flatter him to grind Siuan’s nose further into the dirt. Bryne, of course, rolls with it, because he’s kind of a shitty person. Let’s recall that Egewne’s New Best Friend among the rebels wanted to imprison Rand for no good reason beyond paranoia and treated the closest thing he had to a daughter like crap because her mother stopped having sex with him.
As far as his forgiven transgression, compared to Gawyn’s & Siuan’s, Bryne went on a mission he believed to be wrong, without the reasons or personal knowledge that justified Gawyn’s & Siuan’s decisions, just because of his romantic interest, and used his position to blackmail Siuan into bonding him as a Warder. Remember, on more than one occasion, bonding is put on a level with sex, including by Siuan in that very incident, citing the displeasure of the soldiers’ wives if they were to be bonded. So Bryne performed the moral equivalent of extorting sex from Siuan in exchange for bringing guards on her otherwise-suicidal mission to save the life of a woman who will treat her like crap until the day they all die. And he’s her favorite. That won’t stop the jerk from cursing when he learns she is raised to the Amyrlin Seat for real.
While I agree with most of your post, I think you're being a bit harsh on Bryne 
Re imprisoning Rand, that is the person there to help defend the Realm / Morgase's person, in a time of rioting etc, being concerned when an Aes Sedai suggests someone is a threat.
Re Elayne, there is a fair bit of talk about how similar she is to her mother, so it is possible more Bryne having issues with the resemblance reminding him of the pain of being exiled.
While I do agree with the sentiment that Egwene treated Siuan / Gawyn unfairly in comparison to Bryne, I think he felt obliged to go to prevent Siaun and Gawyn dying, and at least made his views on the overall enterprise clear.
While at times bonding is compared to sex, the view seems to shift a lot depending on the view of the Aes Sedai at the time, I am sure there was an example where they said a young Aes Sedai had a certain warder in mind, and in punishment for her behaviour they forced her to bond someone else while the one she wanted was bonded by someone else again - if the comparison to sex holds true all the time, it paints an interesting picture of what occurred in this instance.
Later on Egwene didn't seem to have much problems with the idea of forcing Aes Sedai to bond warders for the Last Battle, and the reaction to Ashaman bonding Aes Sedai against their will seemed more about the power imbalance than impropriety as such.
