False Premise
Surely the Jedi knew the Republic was a democracy in name only
This isn't correct. It was a democracy. A corrupt democracy at times, sure, but a democracy nonetheless.
The fact that it was so inept in its handling of Naboo during the Trade dispute of Episode I and was so easily manipulated by <PERSON> doesn't by itself make it worse than any other form of government.
In Episode II, <PERSON> even tells <PERSON> that the government's biggest problem was disagreement, not corruption or structure as you imply:
<PERSON>: You really don't like politicians, do you?
<PERSON>: I like two or three, but I'm not really sure about one of them. (smiling) I don't think the system works.
<PERSON>: How would you have it work?
<PERSON>: We need a system where the politicians sit down and discuss the problems, agree what's in the best interests of all the people, and then do it.
<PERSON>: That is exactly what we do. The trouble is that people don't always agree. In fact, they hardly ever do.
Episode II: Attack of the Clones Script
The Republic's Ideals
The Jedi were firm believers in the Republic's democratic ideals, ideals many real-life people share. 503
In Episode III, <PERSON> echos that sentiment saying:
<PERSON>, my allegiance is to the Republic ... to democracy!
Episode III: Revenge of the Sith Script
I feel like I shouldn't need to spend time arguing the virtues of democracy so I'll just say it's generally accepted that democracy = good and dictatorship = bad.
20/20 Hindsight
Despite our feelings on how good of a government The Republic was, we cannot argue with history. According to <PERSON> in Episode II:
There hasn't been a full-scale war since the formation of the Republic.
Episode II: Attack of the Clones Script
That's 1000 years of relatively little conflict. 1000 years. In a fictional universe characterized by the title, STAR WARS, I'd say that's pretty darn successful.
By comparison, the Old Republic saw Sith Wars, the Mandalorian War and more. The Galactic Empire was so poorly run, it only lasted 25 years or so. Now, with new movies and books rolling out, we're seeing a New Republic that after only 30 years in place has had its capitol completely destroyed and new conflict raging.
History tells us that The Republic is the most successful government the galaxy far, far away has ever seen.
It's hard to fault the Jedi for fighting to preserve it. 503