[The Might]

I am not sure myself, but perhaps infromation can be found in the HoMe series seeing how the story evolved.
Of course the story could have lived without it, you see that in the movies where the really important moment that takes place during this chapter - the Hobbits receiving the enchanted blades - is simply replaced with Aragorn giving them these swords.
As Tom Bombadil is not really of M-e it isn't really a big deal that they don't see him and nothing else in this time has great influence later.