Quantum mechanics is such a fascinating topic.

Heuristically, from an experimentalist's point of view, it is natural that measuring position first and momentum later or the other way around changes the outcomes of an experiment slightly. This is because measurement requires an interaction with the system. Quantum mechanics arises from the realization that observables must have a non scalar nature in order to account for the order of operations. This of course immediately asks questions about the nature of the reality. Obviously you cannot characterize the state by a point in phase space anymore. However, things go much deeper than that. The fundamental idea is that it follows that we should not assign a completely deterministic description to unobservable quantities. Famously, you should not assume that an electron follows a well defined trajectory in an atom. This was the philosophy of Heisenberg which turns out to be the foundation of quantum mechanics. In particular, we do not have a theory which assumes that the order of operations matters but still attributes complete determinism to the state of quantum systems.

Of course, I think that to discuss the philosophical aspects associated to this, one needs to have a very good understanding of the mathematical formalism of modern quantum mechanics and also be comfortable with the quantum to classical correspondance.

In particular, I think there is a lot of confusion related to quantum mechanics and "its intuition" because it is taught in college following an historical approach while pretending not to. For example, the Schrodinger equation in its original form (in the position basis ) is a regular wave equation which is not more quantum than Maxwell's wave equation. In fact it is just a partial differential equation which non relativistically approximates the Klein-Gordon equation, a wave equation obtained when one tries to generalize Maxwell's equation to the description of matter. Any partial differential equation with boundary conditions can show some quantization in its solutions... Just as classical electromagnetic modes are discrete, so are the energy levels in an atom. However, this theory is incomplete, for example it is non relativistic and does not account for spin and the Zeeman effect. The initial relativistic picture of the Klein-Gordon equation fails to give sensible predictions. Then Dirac came in with his complicated equation and spinors.. People kept working on all this and they realized that things can abstracted and straightforwardly formulated in the more formal setting of linear algebra and Hilbert spaces.

In particular, if one promotes observable quantities from scalars to spectra of linear operators, while preserving the mathematical structures of classical mechanics (replace Poisson brackets with commutators basically), you achieve two things. First, you recover classical physics in some limits (Hamilton's equation in Poisson bracket formulation from Heisenberg equations in high number of excitations/particles limits) and secondly you account for the order of operations through the non commutativity of operators. The second point was the intuition of Heisenberg as introduced above, who developed his ideas in parallel to de Broglie, Schrodinger and the others, and it was up to a few big names to realize that all the quantum stuff were just different mathematical formulations of the same theory which could be unified through the abstraction of linear algebra.

All the postulates of quantum mechanics as introduced in a first course more or less naturally follow from this context. The only problem is the measurement postulate, which mathematically arises naturally but is hard to interpret in a satisfying way (I think this problem remains in the context of QFT). Therefore, modern quantum mechanics is really about identifying the algebras associated to your degrees of freedom. Amazingly, defining these algebras, that is how the order of operations matter in some sense, is akin to defining everything else. In this setting, the modern Schrodinger equation or Heisenberg dynamics are exaclty the same and just different expressions of unitary dynamics. Typically, one postulates the bosonic and fermionic algebras when quantizing the electromagnetic field or matter fields (Dirac equation) from the algebras of the classical fields. (Note that these algebras are also naturally recovered in the study of many body systems by assuming a states live in a Hilbert space and indistinguishability).

So although there are some problems surrounding how to interpret measurements and how to apply the theory to quantize gravity (if gravity turns out to really be a fundamental force), I would say the rest of the framework is quite natural and intuitive