The Weyl-Wigner representation is a useful tool to study QM from a semiclassical, phase-space point of view. My question is simple: if this method is so close to classical mechanics, why don't we use it for the whole theory of QM? Why do we need Hilbert spaces?
Wrong? There is nothing "wrong".
The phase-space formulation of QM, for which the Wigner-Weyl ordering prescription is a particularly popular flavor, is, indeed, completely equivalent to the Hilbert-space, and also the path-integral formulations.
Equivalence of results does not dictate equivalence of ease, techniques, and convenience. Ask the same question for the path-integral formulation as compared to the majority operators-in-Hilbert-space one. In most of the things you use QM for, solid state, atomic physics, etc..., you use the latter one. But you use the former one to simulate QCD on the lattice, or introduce and derive functional equations, ghosts, anomalies, etc. Each language has its advantages and disadvantages.
It is hellish to do the hydrogen atom in the phase-space formulation, especially on first principles, but it has been done, and has furnished valuable insights (but mostly for understanding pathologies of the Bohr atom). Many techniques, including perturbation theory, are messy in this formulation: the differential equations involved are of infinite order, in general; but, of course, as you seem aware, appreciation of the classical limit of QM is much easier and more natural here, as you end up contrasting apples to apples, and not oranges.
But classical mechanics techniques are not that easier than QM ones. In fact, people resort to Hilbert space to study special features of classical mechanics (Koopman--von Neumann formulation.)
If you speak several languages, there is also a bevy of things easier to express in a particular language rather than another. Within Hilbert space QM, even, people use the wave or matrix formulation in different problems, or the Schroedinger/Heisenberg/Interaction pictures, alternatively, even though they are all equivalent.
In fact, within the phase-space formulation, people opt for different flavors of it for different problems, like the Wigner-Weyl you mentioned, the Husimi, the standard or anti standard ordering, the Born-Jordan ordering, etc.
The $\star$-product is highly non-trivial except for HW. Even for $SU(2)$ this is quite complicated. And without a $\star$-product you are reduced to pictures or to the TWA, which only does so much. In other words, unless you are working with quite simple “flat” phase spaces where the $\star$-product is “easy”, you are in for quite a lot of technical difficulties.
Klimov, A.B. and Espinoza, P., 2002. Moyal-like form of the star product for generalized SU (2) Stratonovich-Weyl symbols. Journal of Physics A: Mathematical and General, 35(40), p.8435.
Martins, A.C.N., Klimov, A.B. and de Guise, H., 2019. Correspondence rules for Wigner functions over SU(3)/U(2), Journal of Physics A: Mathematical and Theoretical, 52(28), p.285202.