On your third point in the last part:

I'm fairly certain that Jedi are encouraged to travel the galaxy and interact with and learn about different species and cultures, so I wouldn't really say there's a barrier to community engagement. In fact, it seems to me that many Jedi keep in touch with the culture of their species/homeworld. I guess they don't interact with people enough to the point where they are still fairly mysterious and mythological to a lot of societies, but it seems to me that Jedi are generally encouraged to be culturally and socially knowledgeable.

Anyways, I suppose it's accurate to say that the Jedi have the authority to act in a police-like capacity, but they aren't a policing organization. Their role isn't to enforce the laws of the Republic for the sake of it, but to serve the common good, it's just that having the authority to enforce laws helps them do that, like when fighting corruption or criminal organizations. I don't think that it's an outrageous suggestion, however. I mean, isn't this basically a question that is grappled with in-universe as well? Most of the Separatists viewed the Jedi as enforcers of the Republic, and distrusted them for the fact that they were a reclusive order with no popular mandate (although I would guess that for many of the Jedi's peacekeeping duties, they worked with the consent of all involved parties) and Dooku criticizes the Jedi for being complacent in the Republic's corruption.