Tom represents the Great Mystery, the great unknown of the beyond that is all seeing and all knowing and all powerful. Omitting him is like omitting the whole premise of the book.
It's like making a movie based on the Bible but omitting the Resurrection of Jesus.
Sorry, but that just doesn't compute for me at all. Tom is a silly character with no gravitas. He puns his own name just to make it rhyme with "yellow" -- over and over again. And there is nothing in Tom's story that is even remotely comparable to the Resurrection. The Resurrection is a key event in the Bible story. It is something that happens to change the dramatic arc of all mankind. In Tom's story, nothing happens at all.
The scene at Mount Doom where the Ring is finally destroyed is analogous to the Resurrection, not Tom Bombadil.
Tom is perhaps silly, but just the fact that he can is seemingly immune to all the Ring can do (with it being implied to be the most powerful artifact in all of Middle Earth, strong enough to even influence Maia like Gandalf) makes him have enough gravitas. If anything, his silly antics make him even more intriguing: how did he gain that power and is it because or despite his silliness?
It's like making a movie based on the Bible but omitting the Resurrection of Jesus.