If they are funded by the Republic (which makes sense), what they are bound to depends on the "contract" they have with the Republic. In any case, no, they are not less beholden to the Force for guidance. The Force is their guide. They don't seek guidance from the senate or the politicians. They are the ones who guide and advise, not the other way around. It's clearly established that the Jedi don't have a say on the politics of the senate. Their role is separate from that. It's also established that the Jedi as a body have a certain independence, something that Palpatine undermines and attacks directly in ROTS.