They didn't get any new evidence on Littlefinger, all they had was 3 teens claiming he did those things and everyone was still cool with LF being murdered. So why not do it in episode 1? Why have all this elaborate fakery?
It wasn't about evidence, what are you on about? They didn't know the Littlefinger's role in all of the events that took place, they know it now that Bran told them.
As for why everyone took their words without any proof -
1) They are Starks and their lords.
2) It all makes sense.
3) Nobody likes Littlefinger anyway.