He never did much with his powers, did he, and now that he's king he's going to have so many demands on his time that he's going to be able to do even less.
Honestly, WTF was Tyrion thinking? Sure, maybe Tyrion had time to find out that Bran wasn't actually crazy, but that's what everyone in King's Landing and all the nobles of Westeros are going to assume about the weird kid in the wheelchair. And they had enough trouble with the last "Mad King".
If Tyrion wanted a do-nothing placeholder for a king, he should have pushed Gendry for the new throne. Really, he'd give the public a much better "story" than Bran, and he'd look handsome and down-to-earth in public appearances. The people would adore him.
I’m sure a weird kid in a wheelchair seems a lot less threatening than a paranoid man who likes to burn people alive.
That is, IF Bran doesn’t reveal to the nobles of Westeros that he know every single shitty little secret they’ve got. THAT could trouble the Lords a lot…
And Tyrion? He simply saw the advantages of a King who knows everything.
The "good story" part … well, I didn’t really get that either. As you say, a bastard from Flea Bottom who fought his way up would’ve worked just as well to me. But maybe Tyrion didn’t want to overstrain things. For thousands of years, they had highborn men for Kings, so maybe the best choice in the transition from "bloodline Kings" to elected Kings was a Highborn.
It's true that there just weren't any good highborn candidates for the role of king available. Still, BRAN? Bran who creeped the fuck out his friends and relatives when he tries to offer sympathy for once? Bran of the flat affect and knowledge of everything that's none of his business? Sure, he wouldnt look personally threatening, but all a king has to do is say is "Burn that man alive" and it's done, and you never know what a man who looks that crazy will command.
He'd be as disaster as a public figure, people who didnt know him well would would find him pathetic or scary. He's the sort of resource a good ruler puts in as nice private as apartment where he can be asked about things, and kept out of public view.
Game of Thrones is not a documentary. People are looking for it to make story sense, not documentary sense. People were not pissed off about arya killing the night king because it was any less logical than Jon Killing him. It made no story sense, because arya had nothing to do with the Night King. In the same way, Bran becoming King makes no story sense.
But, but, but did it SuBvErT YoUr ExPeCtAtIoNs!!??