Quantization and Commutation Relations

In response to your reference request, phase space formulation; path integrals. Perhaps this should be reassigned to community status resource recommendation. If full books were required, in your shoes I might well opt for CTQMPS, and Feynman & Hibbs respectively.

No, the Hamiltonian in a path integral is a c-number! Admittedly, the recipe through which you actually evaluate infinitesimal shifts and averages does pick ordering prescriptions (L. Cohen 1970), and thus some non-commutativity prescription in the conceptual background.

(Analogously, the star-products of deformation quantization determine telltale star-commutation relations. Quantization is a mystery.) So it's fair to think of quantization and the commutation relation joined at the hip, but the narrative of the 1920s you are considering fits in a much-much broader landscape.

My original comment eliciting the resource request was

"Why" is an impossibly broad question. It just happened that Heisenberg discovered non commuting matrices that did the trick (described quantum radiation) and his boss, Born, understood these obeyed commutation relations. In another universe/planet, they could have deformed classical variables by pseudo-differential operators, to describe the same systems. In yet another planet, Dirac/Feynman could have hit upon path integrals to do the same job.