Jaime was spared when he killed the King and that turned out awful for the realm. I'd say they wisely did not want a repeat of that.
I'd also guess that after murdering the second woman he was ever in love with and watching Kings Landing be decimated he probably did not want to stick around. Him returning to the place he called Home for the past 5 years actually makes a lot of sense.
I agree about Tyrion, but who is to say that is not going to end in disaster. A cripple king, a hated dwarf as Hand, a female Commander of the Kingsguard and two up jumped peasants on the Small Council is a recipe for disaster. I give it two summers before the first rebellion.
It would have made more sense if that Council picked her to be Queen given Robin, Royce and, presumably the other Vale Lord, would have voted for her, Edmure, already intimidated by her Cat impression, would have done so as would have Tyrion and the other Starks if they got a vote.
Bran makes little sense given there is no real evidence to most of Westeros of why he is so special or that there realm was in any real threat given all the Others simply exploded. Sansa being Queen would have meant the North remaining.