In the movies, the Jedi always seemed to expect everybody to believe in the Force.
Where in the movies is that shown? From what I saw, it seemed the Jedi Order as a whole and single Jedi couldn't care less if anyone among the general public believed in the Force or not. They themselvees know it is an energy field that exists in the universe and effects everyone, whether they choose to believe in it or not.
Kind of like you could say you didn't believe in gravity and walked of a cliff; gravity wouldn't care one bit in any of your beliefs and drag you down anyway.
Neither to the Jedi, nor to the galactic citizen, is the Force some kind of religion - with the possible exception of Admiral Motti, who called it an ancient religion once. And unlike any religion in real life that blasphemy actually had an immediate supernatural reaction from one of its practitioniers.
As @EikePierstorff stated in the comments above: If I walked up to a Norse peganist and said I didn't believe in that storm god Thor and if he was really a god he would struck me down with lightning from the heavens right now... well nothing would happen... or if it did, almost anyone would convert to that deity pretty quickly.
They also wanted the Republic to trust their powers and rely on them.
Eh, no. The Republic made them their guardians of peace, ambassadors of good will, whatever you want to call it. The Jedi themselves exist solely to serve the will of The Force.
However, they only trained a few to become Jedi. They also didn't share the knowledge of the Force.
History has shown the Jedi Order that knowledge of the Force in the wrong hands mind-boggingly often leads to a bit of trouble (aka galaxywide conflicts with billions of sentients getting killed), so they got understandably vary who they entrust with their knowledge.
Why did the Jedi preach in an uninspired way?
Because they were never actually preaching. They are not trying to convert anyone. The Jedi Order is not a religious order, nor a cult. A person can't learn to become Force-sensitive after all. It's like being magical in Harry Potter, you're either born with it, or you aren't. You're either chosen to be part of this other world, or you aren't.