We can certainly imagine part of what the GP comment described being true: "a more fundamental substrate to reality than fields that fields emerge from." In fact, many physicists assume that's the case with the Standard Model - that e.g. the similarities between generations of quantum particle are explained by some deeper and (hopefully) simpler construct.
Similarly, "Maybe the N fields are just vibrational modes or attractor dynamics of something simpler" could also be true - Calibi-Yau manifolds in string theory are essentially one such attempt to unify the similar and repetitive aspects of QFT that currently have no theoretically-justified connection in the theory.
Sure, at some level you presumably hit a wall - e.g. "why are there Calabi Yau manifolds?" But I don't think that's what the GP was referring to.
> Any such discovery is contingent upon some arbitrary choice of axioms.
This is true, but we see some wonderful examples of this in the real universe, producing laws that must be true in all universes that satisfy the axioms (assuming we believe that mathematical proofs aren't somehow tied to our universe.)
For example, Noether's theorem tells us mathematically when and why conservation laws, like conservation of energy and momentum, exist (i.e., for any continuous symmetry of the action of a physical system with conservative forces.)
Similarly, the inverse square law applies to anything that propagates, with no losses, outwards from a point in all directions in locally flat three-dimensional space. Again, we expect this to be true in any universe with these properties.
There are quite a number of other examples of this.