Wow, It did not ring true to his character. I do not see him pathetically begging like he did. I imagine he would have kept his composure a d gone out refusing to show any weakness. For a man who has had such control over everything, went out in such a weak way, and not a poetic weak like Tywin going out while on the toilet.
Yeah, I agree with that, that whole branch of the plot is rotten. But, so much for due process in Westeros. What was going on in that room, if they were going to kill him just kill him, don't do this poor idea of a kangaroo court. And Littlefinger is a pragmatist, he must be used to the idea of and accepted the risk of his life choices ... he got found out.
The kangaroo court was necessary: Sansa had to demonstrate that killing Littlefinger wasn't an arbitrary action. And if she hadn't caught him by surprise, Littlefinger would have had time and space for escaping, and now becoming a very dangerous enemy. Witnesses had to hear him either confess or successfully declare his innocence. Otherwise they could have characterized Sansa as another Cersei, not worth trusting or swearing loyalty to.
He wasn't as smart as he thought he was and then got too cocky on top of it. He had a lot of power for a longtime. Plus, we as an audience had only seen Littlefinger as the controller and manipulator so when we finally saw him as the person being manipulated, of course he is going to act differently.