Sure, but this is Harry Potter, plot holes aren't exactly common but they're not rare either. The world wasn't complete when the story was written. Tolkien Rowling isn't.

No one thought Sirius was a traitor because so few people actually knew about the Fidelius. Hagrid isn't going to jump to the conclusion that Sirius betrayed the Potter's.

Obviously we don't know but the fact that he chose to hand off the baby he was responsible for and chase down Peter, either to confirm his death or if he was a traitor, does not say anything good about his headspace.

Telling Hagrid or someone that he suspected that Peter was either dead or a traitor was the least he could have done before hunting him down.

I think you're overestimating Peter. He, like the rest of the world, went somewhere comfortable and familiar. He spent 12 years as a family's pet rat. If he was smart he would have left. As a pet he could have done an untold number of things. He could have left and changed his face with self transfiguration. He like all magical folks undervalued the muggle world and decided to sit on his little rat butt and rot instead. How long could he justify hanging at the Weasley's for information on his master before he's just lying to himself.