It's neither Etzel.

By the end of TGS before entering the Hall, Egwene essentially told Gawyn he needed to reflect on their relationship and understand what it would mean to be her lover and warder, and reflected on his errors and amended himself before he committed himself and she agreed to bond him. As Amyrlin she expects his obedience when she gives orders, and he went against her authority as Amyrlin. She won't have that, and she won't bond Gawyn unless and until he corrects that. Gawyn has to end his inner conflicts and choose where he stands and stick to what he committed himself to. He can't be Elayne's First Prince one day, want to give it all up to go live a "simple life" with Egwene on his country estates (which he offered her to do in LOC) the next say, side with Elaida and lead young warders-to-be
into a political conflict only than abandon his men and his commitment to the Tower to go to Egwene's side later (the Younglings is another issue Egwene has yet to decide what to do with... and considering what she did with the Rebels Aes Sedai, what she choose to do with students who stepped out of line and killed warders might not be pleasant. How would Gawyn react if Egwene decides the Younglings can't come back?), only to disregard her authority almost right away. Gawyn right now is very unreliable. Galad may follow his moral radar blindly (though Elayne makes it worse than it really is), but Gawyn is childish and listens way too much to his emotions and shifts with the wind. Morgase in normal times would have been furious that a Prince of Andor took side in the Tower Coup, for instance. That was most undiplomatic and completely out of place - had Elaida lost her gamble, the repercussions to Andor could have been serious, with Gawyn having fought to topple the rightful Amyrlin Seat.... Gawyn was totally out of line - it was the Queen's decision to show her displeasure with Siuan or not and to what extent, not Gawyn's.

The fate of the Egwene and Gawyn relationship was uncertain in the short term at the end of TGS, though it's most likely a bump on the road. Egwene won't bond him casually a few days after, and he certainly won't be bonded the next time we see him. Egwene is expecting some character development from Gawyn before she makes her final decision.

In the short term, Gawyn was quite put off by Egwene's reaction, and it's not certain he will react as she wants him to react. If he waits too long before showing signs of growing up or makes a firm commitment, she may very well send him back to Elayne in the meantime, or he may react childishly to what Egwene told him and announce since she won't have him he's decided to return to Elayne and his duty in Andor.

IMO, it's likely Egwene won't bond Gawyn before the "big event that launches TG" at the end of the book. If she does. Gawyn also has a decision to make. Until the end of TGS, he never realised that a relationship with Egwene meant being the warder not of any Aes Sedai who could follow him back to Andor but to be the warder of the Amyrlin Seat. As her warder, his duty would be at her side, not Elayne's. Egwene won't have a warder who doubles up as the general of a nation (that would be the end of the appearance of neutrality of TarValon....), and Elayne won't have a First Prince who's in Tar Valon and not at her side where he belongs. Gawyn didn't know until recently he had conflicting commitments. He needs to make his final choice between Elayne and Egwene. Before TGS it was obvious he would choose Egwene, but it's no longer sure their relationship will continue. Bryne even tried to make Gawyn see that his duty was to Elayne's and not at the Tower, that he was sworn to protect her and become her First Prince.

Egwene's coldly rational reaction to Gawyn and his behaviour in TGS might actual be the beginning of the end of their relationship because duty separates them, not the last step before she bonds him and they marry. Egwene has left quite a few of her romantic notions about being the Amyrlin behind through the Tower conflict.

Or course, there's also Elayne in this. She was all in favour of Egwene and Gawyn as a couple as a bleary-eyed Accepted whose mother was expected to rule for decades still, but now she's Queen and during the civil war in Andor, she was fully expecting and hoping Gawyn would return to take up his duty soon. Because of her own decisions concerning Rand she's not really in a position to tell Gawyn that duty calls him away from Egwene, but still... Elayne didn't mix her personal feelings and politics when it came to making her decisions about Rand and Andor. She acted as a politician, even though as a lover she'd rather not have done that. As a sister, she might bless his relationship with Egwene, but as Queen will she?