Since Mo is a Sedai, if she just swore to not come back unless Siuan allowed her to, then she would be forced to do that, right?
If she swore under the Light and hope of salvation and rebirth, she wouldn't break it regardless of Aes Sedai - nobody would. I don't think Aes Sedai are forced to hold to their promises though, however she could obviously not say it unless she meant it. Make a promise that you intend to keep, and if you genuinely change your mind later because of something, I doubt it would hold you.
I know changes have to be made for reasons but leaving Mat behind just doesn't make sense at all...
I think that the only character that doesn't have a lot of depth right now is Rand. His mother died, yes, but he didn't know her that much, and he had a kind father. I am hoping we get more depth from Rand, he doesn't even have that conflict of Tam's fever dream.
I know that likely he was left behind for real-life actor leaving reasons.
But I think it made sense story-wise as well for Mat as a character to a certain extend.
He has always run away from being a hero and avoided responsibility. Maybe he'd feel sorry for leaving them in the second season and they'd use this for his character development ...