Sorry if this sounds at all like pretentious armchair theorizing(it basically is), but I have a question which I can't really figure out how to search. So, given the resemblance of the cerebellum to the brain as a whole, is it possible that the growth of the brain beyond the cerebellum arose from a mutation forcing the growth of a second, "extra" cerebellum? Again, feel free to downvote this into oblivion if it's either obvious or stupid. Sorry again.
I don't have a clear and definitive answer to give to you but first you should have a look about brain development (from:https://www.apa.org/education/k12/brain-function):
And to the function associated to cerebellum:
And the number of neurons:
And maybe have a look about difference between species :
It appear that cerebellum have specific function interacting with the rest of the brain. It present size variations between species from different branch of evolution....
Hagfish, the most primitive vertebrates, do not appear to have a cerebellum, or if they do have one it is very primitive. Hagfish have most of the other structures found in vertebrate brains (including the telencephalon), so it is likely that the cerebellum actually evolved after the other five main brain regions.
It is possible that the dorsal cochlear nucleus, a region on the dorso-lateral surface of the mammalian brainstem, started out as an 'extra cerebellum'. Some fish and amphibians also have 'cerebellum like structures', such as the dorsal octavolateral nucleus. However, the cerebrum almost certainly did not evolve from the cerebellum.