Appears in narrative for no reason- chapters have no bearing on subsequent events- ring has no effect on him- creates awkward plot hole Tolkien tries to talk around in the Council of Elrond- refuses to elaborate- leaves

I think he's an author insert. The way people talk to and about him is similar to the way people react to Tommy Wiseau's character Johnny in The Room:

It's a pretty dumb scene that occurs when the hobbits are alone in the wilderness. A bunch of shit happens to them so you go "man this middle earth place is CRAZY" and so there's some suspense so it doesn't just seem like they're taking a stroll through the woods to their next meeting with the competent characters. Bombadil was just a convenient deus ex machina, and Tolkien waving him off later like "oh that guy, no don't worry about him, he's never going to show his face here again" is really the only good thing he could have done with him. Other than he could have just edited out the bit about him not being affected by the ring and then no one would even think about him again.

Also: Faramir wasn't affected by the ring either (in the books not in the movie) and nobody is sucking his peepee or saying he should have saved the world.

He exists to filter plebs