The world would certainly be a poorer place without mystery! In a letter of 1954 Tolkien wrote:
...even in a mythological Age there must be some enigmas, as there always are. Tom Bombadil is one (intentionally).
So our feelings are spot on with Tolkien's there!
This is the comment about Bombadil, made by Elrond, that intrigues me most:
"Power to defy our Enemy is not in him, unless such power is in the earth itself"
That pretty much confirms that Bombadil is something totally different from Gandalf, any other Maiar or the Valar, who do have power to defy the Enemy. But there is a lot of belief out there that he was a Maiar because what else could he be? Was he always something different? Did he
become something different because he has always been in Middle-earth from the beginning?
I don't think this passage is saying he doesn't have power, but that it's not power as we know it. He is totally beyond the struggle between Good and Evil that is being played out in Middle-earth in the same way the earth itself is. He is what he is and nothing changes him. Because of that the Ring has no meaning for him. It really is intriguing! To create a being that is outside all that is going on - even the Valar are involved (they sent Gandalf) - is pretty unique I think.
I think this is a very telling passage too:
'Fair lady!' said Frodo again after a while. 'Tell me, if my asking does not seem foolish, who is Tom Bombadil?'
'He is,' said Goldberry, staying her swift movements and smiling.
Frodo looked at her questioningly. 'He is, as you have seen him,' she said in answer to his look. 'He is Master of wood, water, and hill.'
It looks like 'He is.' is the whole answer, although Goldberry obligingly tries to elaborate for Frodo. That's a statement to stop you in your tracks, although it's clear Tolkien doesn't mean us to take Bobmbadil for God. But he certainly shares at least the characteristic of just
being with Him.