The fall of the Jedi began long before the outbreak of the Clone Wars. Becoming behold to the Senate must have certainly seemed like the reasonable and responsible thing to do rather than running around being a law unto themselves. However, the downside of that increased accountability is that they're also subject to the mandates of a *massive* political bureaucracy.
For the Order, the Clone Wars was (by design) a trap wrapped in a Xanatos Gambit, stuffed inside the Kobayashi Maru. The Jedi became Generals in the war in the hopes of ending it as swiftly as possible. A war BTW which they had every reason to believe had been orchestrated by one of their own who had joined the Sith; which makes it very very hard to just ignore.
If they refused, they would 1) probably violating whatever concord they had with the senate, most likely loosing any official authority they might have, 2) be guilty of standing by while half the galaxy burned in an unending civil war. They'd essentially become outlaws and pariahs. Not a move to be taken lightly.
The ROTS novel does a great job of saying that by fighting in the war at all the Jedi lost. But I don't know if they ever recognized why they lost.
I know people hold that novelization in high regard (and rightly so) but there's one aspect of it that's always bothered me; Anakin and Obi-Wan being propaganda celebrities.
Aside from all the problems it creates for the OT, it just doesn't sit right. I forget where I heard it (probably a BTS featurette for TCW) but I recall Filloni saying that George's idea was that Palpatine (who controlled the media) deliberately kept the Jedi out of Republic propaganda, at least in any specificity. Only ever portraying them as faceless and monolithic. Which the Jedi would take as a kindness since their selfless precepts make them uncomfortable with the spotlight, while in reality it allowed Palpatine to further mystify the Order (and not in a good way) while subtly shift the blame for the war in their direction.
I think there's even an episode on Coruscant the features a giant video wall of Palpatine playing the the background where he's basically "publicly dispelling accusations of Jedi warmongering" which seems to support this tactic. It keeps Palpatine the face the the Emp-I mean the democratic bastion that is the Republic, while using the Streisand Effect to erode trust in the Jedi.