In the pages of The Lord of the Rings it becomes clear that the Valar themselves have no idea where the Hobbits came from. Which means that they somehow appeared in Middle Earth despite not being part of the Song of Illuvatar, because if they had been foretold in the song then the gods would have known about them because they helped sing the song.
This makes perfect sense if Hobbits are supposed to be an offshoot of humans -- as the song foretold the awakening of Men in Hildorien the Hobbits would have been technically foretold as well, even if they weren't yet separate enough to be specified. So their close relation to humans concealed their existence from all the gods, even from Olórin, until the time was right.
More importantly their existence was unknown to Morgoth. If they existed in the First Age, he could not target them as he targeted elves. Most importantly their existence was unknown to Sauron, who, like Olórin and every other Maiar, was there to hear the song of Illuvatar, and knew of the coming of Elves and Men. If they were a humble people, set in a land far from the big comings and goings and doings of the world, a land Sauron was sure his captain had depopulated and despoiled, such that he would not bother to look in that direction nor hear of any challenge from it, why then, he would not notice them at all!
Now one could argue that this was not a deliberate effort on Illuvatar's part to conceal their existence. It is entirely possible that they arose entirely by accident. And yet -- unknown to Illuvatar? The One who sang every bit of existence into being? When every single creature in all of Middle Earth was supposedly planned ahead of time? Could they possibly have arisen by accident in a world made of careful planning?
Perhaps. It happens. Perhaps even for Illuvatar's plans. It would explain why they act so differently from all the other peoples -- they were never granted any great Fate so they never had to act like it. No fighting for more territory, no seeking adventure, no dynastic struggles, no challenge to the mighty nor scourge upon the weak, none of that nonsense, thank you very much. Just a cup of tea and a few biscuits. All these big people rushing down the road like they have something important to do, what on earth could possess them?
Or it could simply be that, if the Shire was founded well after Angmar fell and well before Sauron returned, and always guarded in secret by the Dunedain rangers, the Hobbits were able to settle themselves without having to resort to great violence, and never needed to produce heroes or get into the kind of trouble that heroes bring, unlike all the big people, and mark my words, some of those people got into quite a bit of trouble indeed.
Either way, if it were true that the Hobbits had arisen somewhere or other in some far-off place in a long-ago time like the Big People did -- well they did, in a way, somewhere between here and there, but honestly, who cares about such things? Elves and Men and Dwarves, but that's their business. Wherever the Hobbits did originate, they clearly didn't care about it enough to remember, so they couldn't tell you even if they wanted to. Might as well have another biscuit and worry about more important things, like mowing the hay in time this year. If any of them start worrying about Big People things, they might run off and get into all sorts of trouble, and miss dinner. And maybe even miss many dinners. Maybe even grow a foot taller like one of the Tooks did. And then you'd be bumping your head on chandeliers. No thank you! Might as well stay home.
As for the more prosaic reason Hobbits don't figure in the tales of the origins of the Big People: They weren't part of the setting when it was first developed or even when it was well-developed. Tolkien came up with them separately from his Middle-Earth tales, and had no intention of putting them in Middle Earth until someone suggested it. So the two settings were developed with completey different tones in mind. The Hobbit is a children’s tale, with a children's tale outlook, scope and tone, and the Hobbits therein are largely satirical -- a gentle satire of middle-class British people as the author knew them. Middle Earth is a grandiose setting whose narrative arc is heroic, tragic and nostalgic, a sweeing story of the decay of all good things and the diminishment of the great peoples from what they were in elder days --
So when Tolkien plants The Shire in Middle Earth, he’s got two different genres in one story, and each barely understands the other, to the point that the grandiose people completely overlook the modest ones like someone opening a door and having to look down to realize there’s a short person in front of them. The result is that two Hobbits are able to sneak right into Sauron’s back garden.
The Hobbits were not written for the narrative that the Silmarillion presented. They are an outside-context problem for everyone whose fate was intertwined with the doom of the Elves. The story of Elves and Men is supposed to be the most important thing about Middle Earth and then here come these weird little chaps who don’t ask for much at all and they wind up saving everyone because nobody noticed them before because they were too busy being Important.
One might say that they’re the creator’s little joke on all the people in The Lord of the Rings who only care about big things.
Including, and especially, the title character.