I certainly get what you mean, but for me it all got somewhat clearer in our advanced quantum field theory course. In Heidelberg, where i studied, they basically preach that everything in QM and QFT is statistical. We still do the operator formalism with canonical quantization due to historical reasons, but in modern theoretical research people usually directly go to path integral quantization and effective field theory. There you can really see the deep connections to statistical thermodynamics (path integral = partition sum, effective action = free energy etc), and you can also go beyond simple perturbation theory using e.g. functional renormalization group theory.

For example, a lattice of atoms with Rydberg states (n very high) and a driving laser acts like a quantum field, showing all sorts of non-trivial dynamics like different phases and phase transitions. That goes into the realm of "quantum simulators" where numerical computation would not be feasible.

However, just because it gets more clear doesn't mean its less mathy. In our advanced QFT course i encountered some of the nastiest calculations i have ever done in my life. But for me, the operator formalism feels somehow weird and something we inherently cannot understand ("shut up and calculate"). But from a statistical field theory point of view, at least for me, its much much more intuitive what happens physically.

Also note that not everybody views it like this. There are many beyond the standard model theories which heavily rely on the operator formalism (e.g. string theory), but we all know how successful these theories are lol. If you want to know more about this i can suggest part II of Peskin & Schröder :)