I will happily mind your disclaimer but there's a critical event that really makes this character truly bizarre. He wears the ring and doesn't give a damn - all of Gandalf, Giladriel, and Sauron himself are affected by the rings. Gandalf and Sauron are possibly on the same "power tier" as Maia (Galadriel is lesser but still psychicly impressive). It's arguable that Gandalf is a false Maiar and actually an avatar of Manwë but Tom's nonchalance with the ring likely puts him above the Maia - Tom could be a Vala but it's really unclear who he'd be (possibly Lórien but its unlikely) due to his relation to Goldberry and disposition ... They're all well known and it's unlikely they could evade detection in Arda... it's possible that Tom Bombadill is Illúvatar himself which lets us handwave a lot of issues but that makes Goldberry even more confusing... also everything more powerful than a Maiar is sworn to (or restrained from) interfering in Arda.

Tolkien was a dedicated world builder and also a myth teller and we might just have to accept the unfulfilling explanation that Tom Bombadill is the mysterious guide that defies explanation and that's all there is to it - his character is completely isolated from the rest of the mythos and this from a nerd who has a detailed lineage of kings for Gondor including the "not at all relevant to the story" kin strife of Castamir... so I reject any claims that Tolkien just "forgot" about Bombadill - the omission of history was clearly intentional.

Anyways, I've spouted off like a nerd long enough. It's an excellent topic to delve into and there are some great hints pointing one way or the other.