Well it works well enough for them. Rather than avail themselves of the system, they decided to be judge and jury. As I said up thread, they'd been in shit if Palpatine hadn't been a Sith. Again, I blame it on weak writing in that part of the movie.
And what if he had walked? What if they people had simply said to the Jedi "STFU, we like it this way". Then what? Would that have the authority to still take him out just cause he's a different religion that them? You have to question is they was really worried about him gaming the system or the fact that their authority might be overruled, that the system might legitimately agree with him.
The reason why the Jedi confronted him was to endure he abdicated his power and, if he didn't willingly, arrest him. It was clearly implied in the film that this was part of the Jedi "constitutional" mandate.
Whether the people wanted to keep it that way afterwords was irrelevant and would be something to vote on later. The Jedi had to follow their immediate legal obligation.
And why couldn't they arrest him as being a Sith? It isn't like some kind of racial profiling thing seeing as how there is an extreme limited
number of them. If he was a Sith, there were pretty good odds (like 50% or better) that he was the ring leader of the separatists; that makes him a traitor. Last time I checked, high treason was a pretty severe capital crime in pretty much any society.
Non the less, right or wrong, once he slaughtered three of his captors and Force Lightninged the fourth, all bets were off.