It's never been said on screen that Starfleet didn't trust Picard post assimilation. In fact, the opposite seems to be true as he was allowed to retain command of the Enterprise and given command of the Enterprise E when the Enterprise D was destroyed. Additionally, he was given several high profile missions after his assimilation including two missions to Romulus (a secret one in Unification and a less secret one in Nemesis-ordered by Janeway), command of the Federation fleet during the Klingon civil war in Redemption, welcoming the Evora to the Federation in Insurrection etc.
I think they did have some reservations about putting him up against the Borg but I don't think they were hesitant to promote him because of it. In Descent, Admiral Nechayev came down hard on Picard for not taking the opportunity to destroy the Borg in I, Borg. It's important to note that she came down on him but didn't punish him, she simply ordered him to take the opportunity in the future should it present itself again. That may explain why they didn't want him at the battle of sector 001 in First Contact. They didn't think he could make the hard decisions when facing the Borg.
Regarding the decision to promote Janeway over Picard, I look more to Generations than First Contact for an explanation because it's really the only thing that's ever been said on screen regarding his promotion prospects.
James T. Kirk: Captain of the Enterprise, huh?
Jean-Luc Picard: That's right.
Kirk: Close to retirement?
Picard: I'm not planning on it.
Kirk: Well let me tell you something. Don't! Don't let them promote you. Don't let them transfer you. Don't let them do anything that takes you off the bridge of that ship, because while you're there... you can make a difference.
I get the feeling that-based on that conversation, coupled with his experience in Tapestry (when he found out what his life would have been if he hadn't commanded a starship), Picard chose to remain a captain. He may have been offered a promotion or two off screen (why wouldn't he?) but turned it down because he's happy being the most prominent captain in the fleet.