Egwene has been the most mature of the Emonds Fielders since the beginning. Since the start of the series she's openly stated that she's wanted to become an Aes Sedai. Nynaeve, Mat, Rand and Perrin all denied and resented their destinies for the greater part of the series. How can you say that everyone has matured except Egwene? If anything, the others are finally starting to behave like adults.

There are multiple quotes in the series showing that Rand doesn't want to trust Egwene. He's paranoid by nature. When she arrived to warn him about the Aes Sedai trying to capture him, he laughs at her and ignores her warning, only to be captured a week later and tortured. By TGS, everyone is afraid of Rand, even his own father. And don't forget... he is insane due to some effect of the taint, and other issues.

Almost all the Emond's Fielders treat Rand the same as they used to. Perrin has nothing to say because he's too dimwitted, but even he has commented. Mat is childish with no real purpose in life unless someone gives it to him, and he too will say something of use. Nynaeve and Egwene are the only two who tell Rand exactly what they think, with Nynaeve doing exactly that in TGS despite his homicidal tendencies. Egwene's certainly going to give him a run for his money when they finally meet. I don't know why you find this surprising or irritating.

Finally, Egwene has obligations as the Amyrlin Seat. She's not going to let the DR enslave the White Tower just because he needs them. He tried that with Tuon and the Borderland armies, and it failed. He needs a new tactic especially against one of the most formidible Amyrlin's in history, who is also notoriously stubborn in his presence.