He declined that... Tolkien said Tom Bombadil had been 'invented'. He considered his wife Edith: to have been Lúthien, so that would make Tolkien - Beren. "I never called Edith Lúthien – but she was the source of the story that in time became the chief part of The Silmarillion."
Tom Bombadil predates the story of the ring. It's placed there as a very powerful creature - still would lose to Sauron if all the world fell apart according to Glorfindel [and Gandalf doesn't object either]. That creature has no direct influence on the story, being more of a bystander... similar to the reader.
I suppose it would come down to if you think the Ring could influence the reader of the story, if it were possible for the reader to actually acquire the Ring. Tbh, it seems obvious to me that any ordinary human would probably fall to its pernicious influence. I certainly would, and its powers are such that I just can't imagine any human could resist its charms. Yet, Bombadil is completely untouched by it.
Tom Bombadil is like the Middle Earth equivalent of a Buddhist sage, just existing without wishing to be more or less than he already is.
Much like Tom, the reader is out of that world, so the ring power doesn't extend... the reader can see the unseen (Frodo/Bilbo having the ring on) and so on.
This is one of those theories that can't really be disproven (like the many eyeroll-inducing theories about the contents of the briefcase in Pulp Fiction) but it also makes little to no sense and is clearly not what the author intended.
Consider:
Perhaps Tom Bombadil represents Santa Claus? After all, Santa Claus has magical powers that utterly transcend what mortal humans can even comprehend. And yet the remains detached from worldly events, never taking a side in wars, never helping human beings during crises that could clearly benefit from his powers, etc.
The truth is that, characters like Tom (and Santa Claus -- if they are indeed not the same person) are fairly common throughout fiction and mythology.
After all, if you invent a character with absolutely transcendental superpowers, you need to explain why the character has chosen not to use those powers to i.e. cure world hunger, or end war, or whatever.
You can see this in... well, any major world religion. Notice how God never directly intervenes in the New Testament, even though he clearly could? Yeah, it's explained.
Tom Bombadil predates the story of the ring. It's placed there as a very powerful creature - still would lose to Sauron if all the world fell apart according to Glorfindel [and Gandalf doesn't object either]. That creature has no direct influence on the story, being more of a bystander... similar to the reader.