I think we're completely talking past each other. All the issues you've mentioned are valid issues but they're about the physics. They are relevant and importa

I think we're completely talking past each other. All the issues you've mentioned are valid issues but they're about the physics. They are relevant and important but, as you say, the Standard Model and QFT have been very successful at describing a large number of physical phenomena at a certain scale, so the model is fine. Its applicability might be limited to certain scales or situations but that's not really different from any other theory in physics. The problem with quantum field theory from a mathematical point of view, however, is that many of the mathematical objects that the symbols in the equations in physics text books represent (e.g. the "quantum fields", operators, inner products, integrals etc.) don't even exist in any (mathematically) meaningful way. Physicists pretend they do, manipulate them following very questionable rules (The limit N → ∞ doesn't exist? Doesn't matter, we'll apply it anyway and commute it with the integral!) and only the fact that, somehow, when the dust settles, those equations produce numbers that we actually see in experiments, saves the theories. Take, e.g. the path integral for any of the field theories in the standard model: Physicists derive predictions from it. Yet it doesn't exist[0]: The integral doesn't exist. The measure doesn't exist. The sigma algebra doesn't exist. Etc. etc. [0]: Or at least the question of its existence has eluded entire armies of mathematicians. So at the very least it is undefined.