Today is Tolkien's birthday, BTW. He would be eleventy-eight today.
Short version: Tom Bombadil fits just fine when he’s not talking or especially, singing.
Even at first, magical reading of LOTR, Tom Bombadil seemed not to fit in the story somehow. The explanation about him made sense enough, and in a middle-earth where trees can have it in for you, a creature who can counter them by talking to them makes sense enough. But the nonsense verse is jarringly bad, not fitting the character at all. His very speech and conversation seem flat, as if some nine-year-old had somehow inhabited the body of some powerful wizard.
It didn’t get better on rereadings, save that I skimmed the dialogue more and focused on the narration on successive passings. The four times I read LOTR out loud to the boys, that section was always tough to put some feeling into.
I had the same impression in Rivendell, less strongly, when the elves sang less serious songs or teased others; this in turn reminded me how the dwarves’ song about Bilbo and the plates at the opening of The Hobbit doesn’t work either. You see the common thread: creatures from Faerie who are both serious and merry, young and old, wise and nonsensical, have one side of them captured in word, but not the other.
In truth, it is hard to do, this putting of words into the mouths of creatures whose otherness is important. Specifically, the light side of that is hard, for Tolkien manages the seriousness and agedness of all these quite well. But this humorous side of them eludes him; I can’t think what other writer does it well. Lewis manages this light side a bit better, though the deep seriousness of some of his characters is less convincing than Tolkien’s.
Bearing this in mind, Tom Bombadil actually works quite well once they have left his house. When he comes to the Barrow-Downs in rescue the verse is still weak, but not so bad. He hears Frodo’s song at some great distance, easily demonstrates great power over things natural and unnatural with just a few words, shows great appreciation for beauty and ancient history, and sets them on their way. Note that as they travel to the Great Road, Tom is described as joking and singing, but no jokes or songs are described. This Bombadil works fine.
Next, Elrond describes him at the council and Gandalf takes up the discussion, with reference to his great age and type of power. Then at the very end of LOTR, Gandalf indicates he is going to visit Tom and have a long talk.
Living at the edge of the great story and out of the track of usual travel, like some strange-named, remote hero out of the Finnish Kalevala, Bombadil makes entire sense. Only when Tolkien attempts to bring him close and speak to us does he fall to pieces. He would have been better-described if the song about him had been one Sam had learned as a hobbit-child, and we heard only evocative snatches, ranging in length from a phrase to a couplet, from Tom himself. The dialogue might still be weaker than other parts of the story, but not glaringly so.
We have not met the real Bombadil – Tolkien doesn’t capture him well.