Turns out he wasn't fired-- he chose to leave. According to this article, he resigned primarily because of conflicts with assistant head coach Lance Reynolds, who just retired, so that problem (if it really was a problem) is essentially gone.
Before I go on, let me say that I still like Brandon Doman. I think he made some mistakes (maybe more than he should have), but I think he did a decent job with what he was given. All-in-all, I think it was Bronco's refusal to play anyone but an injured Riley Nelson at QB and the inability of the O-line to protect Riley from getting hurt again and again that really prevented the offense from being productive during the tough games of this past season.
Yes, Brandon made some really bad calls (for example, the "miscommunication" that caused Taysom Hill's season-ending knee injury), and while it was probably enough for him to deserve to lose his job as OC, it definitely wasn't enough to have him be run out of town with flaming pitchforks. For that reason, I think it was a very smart decision on Bronco's part to offer Doman the QB coach job-- showing he still trusts him and respects him, but that he also believes, rightfully, that changes need to be made.
I wouldn't blame Doman if he decides to turn down the offer of QB coach and apply to be the OC somewhere else, but I hope he decides to stay at BYU. I think he's a good fit for the team, and I think he would be better off focusing on just the quarterbacks instead of trying to run the entire offense, at least until he gets a few more years under his belt. But if Brandon does decide to turn it down, I hope Max Hall gets the job. He'd be amazing!
Being the offensive coordinator really is a "thankless" job -- it's impossible to please everyone, and OCs have to be able to withstand a lot of criticism and be constantly trying to improve, against the odds. Doman just wasn't up for it. Anae, on the other hand, knows what it takes, and he's proven capable of handling the pressure.
Last time he was at BYU, Anae worked with Max Hall, Austin Collie, Harvey Unga, and many others who have since become legends. He helped them pull off some really spectacular wins. This time around, he'll get Cody Hoffman, Jamaal Williams, Taysom Hill, Ross Apo, and a few others who have all the talent and potential, but need to find ways for it to be used to its fullest-- I think Anae has a pretty good shot at helping them shine. We'll find out in September!