sheepy_dogs_handThe Starks are obsessed with fair trials and their code of honour. If Littlefinger was accused, he would ask for evidence and a fair trial which would have to be given (or a trial by combat). Winterfell is the last place in the country that would suddenly lose its legal system.
Executing Littlefinger without a trial based on evidence that consisted of "bran had Magic visions" would have made Sansa look like a complete tyrant to all the nobles who were present and the people would have been shocked, especially considering he saved them all with his army and was one of the most powerful people in the country. .
I've never read the books, and I don't think they should play too deeply into discussions about the show. Not saying you can't prefer the books over the show, but it seems like every discussion I come across devolves into complaining about how different the two are. This thread is about the TV series, so I'd like to discuss that instead.
Since when are Sansa and Arya obsessed with fair trials? They've both spent the last several years experiencing some of the worst that humanity has to offer. Their last experience with Stark honor was watching their father pointlessly die for it. Arya certainly would feel no qualms about killing without a trial at this point, and Sansa has put up with Littlefinger's shit for so long, I imagine she's been dreaming of killing him for years. As for how the rest of the lords at Winterfell would feel about execution without a classically fair trial? Like I said, everyone at Winterfell has had to very quickly accept the fact that magical, unexplainable events are about to play a huge role in their lives and deaths, and need to adapt to this new reality. At one point, Winterfell might have been the last place to abandon it's traditional legal system, but now it's going to be the first city to fall to the white walkers, so necessity dictates otherwise. It's not difficult at all to demonstrate to any man the extent of Bran's sight, and right now the top priority is to consolidate power; to cut out any weak links that might make fighting the undead any more difficult. Winter's coming, winds are blowing, et cetera, and I think any man in the position of the lords at Winterfell would just be grateful to have a strong leader whose word they can follow. Doesn't seem far fetched to me that they'd be desperate enough to just be happy they've got some magical all-seeing mage on their side. That, or they'd fear him enough that they wouldn't dream of revolting. Either way, they presumably all believe in the threat, and the last thing any of them are going to do is rock the boat during a hurricane.