Here you are speaking using words like "electron", "thin barrier", "oxide" and so on. So you are thinking in a framework of our culture and our variant of STEM. Doing so you are trying to prove that there is no way to invent another variant of STEM. I believe, that such an approach could achieve nothing.
I mean, to speak "electron" you need to rely implicitly on a whole mountain of theories about reality. So even if you proved that one cannot build MOSFET without QM, it would mean that he cannot do it while relying on this mountain of theories. It wouldn't prove that there are no other mountains allowing to invent MOSFET.
> Can it be different, can you do QM without Schrödinger Equation? Sure, but it will still be QM.
It is a trivial proposition. "QM without Schrödinger Equation" is QM by definition. The question is if it possible to build some other models of reality that allow to invent MOSFET. As I see it you are trying to remove QM from the mountain of theories and to think what could replace QM. It is completely possible that nothing but QM could replace QM in such a setup. But what would happen if you remove all of the mountain of theories and start from scratch?
I do not know. To see what would happen I need to design an empirical research that would involve some millions of people and a few millennia. To be sure I would need to repeat this at least 50 times.