Umm they didn't outsmart him. Bran the all seer told them the truth. It's not that hard to trap someone when a character in the story effectively knows the script. And even with having the three eyed raven, their trap was pretty weak. They absolutely failed to get any hard evidence against him.
How can you Lords trust Sansa when she said Lyssa committed suicide?
How can you Lords trust Sansa when she betrayed her own brother Robb?
How can you Lords trust Bran when he was not a witness to any of the events he described?
If I can figure out these 3 basic questions that would have shredded the accusations against him then not sure how LF or any character with a brain couldn't have. Now maybe they still kill him because frankly no one really likes the dude but it's piss poor writing that this guy that outwitted so many characters somehow is incapable of making a logical defense in the face of unprovable accusations.
The writers simply rushed things and gambled that most won't care as long as LF was killed. Just like they gambled no one would give a shit about the warp speed from last show because Dragons and the NK. They are sacrificing logic for entertainment which is fine, it's a show afterall but I appreciate smart writing and this was not it.
It's funny how people can watch the same show and come away with such different interpretations of what they just seen. I will grant you the writing left a lot for the viewers to figure out on their own, but still, I thought this episode was one of the best writing wise of the entire season. And that is not to be confused with the most entertaining (aka the special effects of the episode 4 and 6).
The WF storyline left the viewer with the task of knowing that a lot went on behind the scenes, I will grant you that. But some of your points I will address:
1) How can you Lords trust Sansa when she said Lyssa committed suicide?
I don't think they particularly liked Lyssa to begin with, and remember, they were also informed that Lyssa had conspired with LF to kill Lord Arryn by poison. They could have bought into the fact that Sansa was just a scared little girl at the time.
2) How can you Lords trust Sansa when she betrayed her own brother Robb?
Same, she did what she had to do to survive.
3) How can you Lords trust Bran when he was not a witness to any of the events he described?
The Bran factor. How hard would it be for Bran to spend a bit of time with anyone who doubts him and show the Lords his recollections are accurate? Not hard at all for Bran to do, but a instance of just this with any of the Lord would have gone a long way it to understanding why they trust his visions.
LF panicked, he was at first denying the allegations, but when Bran brought up what LF said to Ned stark when he betrayed him verbatim LF lost his composure. At that moment, he knew Bran could dispute, and disprove anything LF might try to bring up in his defense (although it most certainly would have prolonged things a bit). When you add to the fact LF had secretly been whispering into all the Lords ears how Sansa should be the ruler of WF for the past several episodes, and then to have that same person accuse you of conspiring against the North, it pretty much leaves you in a very vulnerable position. One must remember it was always LF intent to install Sansa as lady of WF, and through proxy, or maybe even marriage, acquire the North in addition to the Vale. LF's plans took a hit first when Jon, not Sansa was elected KitN, which lead LF to work behind the scenes to discredit Jon, and then got really complicated upon Bran and Arya's arrivals. LF then started another one of his elaborate schemes, which was to pit Arya against Sansa, which would have lead to Sansa eventually passing judgment on Arya, and thus removing another one of LF's obstacles in his grand scheme.
