Well, the question is about motivations, and different people may gave different ones, so here is what seems to me reasonable.

First about: what quantization is? Imho, texts on geometric quantization mainly follow old-style way of thinking which is somewhat not the best one, and moreover, imho, misleading.
For me it is much more transparent to think of deformation quantization perspective.

So quantization: you have a Poisson (symplectic) manifold and algebra of functions on it.
You know that Poisson bracket is "first-order" of non-commutative product,
so first goal of deformation quantization is to construct non-commutative algebra from Poisson algebra.

Next goal might be -- try to construct its representations.

**Basic belief:** if manifold is symplectic then corresponding algebra has ONLY ONE irreducible representation. (If Poisson, then: irreps <-> symplectic leaves (i.e. orbit method MO).

So I would not say " we always assume that the Hilbert space should not be too big",
but I would say we want to solve very concrete problem: to construct this unique irreducible representation. And geometric quantization is recipe how to do it,
which sometimes work, and many times has problems, which probably can be solved by somebody in some future, but may be some of them are unsolvable in this framework.

In geometric quantization texts, it seems to me expositions does not follow this root.

Moreover - the main mistake which some geometric quantizers make is -- that they want to take algebra of functions with Poisson bracket as Lie algebra and look for its irreps.
This is absolutely misleading. It is not natural, and it is not what people do in physics
even in the simplest example of $p,q$ and canonical quantization.
It leads to artificial no-goes.