There are other foundations for mathematics. Dependent Type Theory has a lot of support from people studying Proof Theory. It is based on typed lambda calculus. It is currently being researched and implemented in mathematical provers, like Coq and Lean.
I understand what you mean, but in my context I would say that you need layers below this. You for example paper, ink, glyphs, and other things before you can start defining ZFC.
You don't need paper or ink to start working with math. You don't need symbols. You can just do it in your head. That's the beauty of math. The physical instantiation of it doesn't matter.
True. But the physical instantiation is necessary. That is what I am exploring: What is necessary to start working with math, and can it be described like the OSI model?