I am currently writing a C compiler, with my own backend (and hopefully, frontend) in OCaml.
But why write your own C frontend? It’s much more of a pain than people imagine. I maintain a C frontend implemented in OCaml (the project itself goes back 25 years) and it’s still not on par with GCC or Clang.
For any other language, sure, but C has so many “wonderful” features, starting with the lexer hack. Your grammar conveniently overlooks this issue but it’s something you’ll have to deal with to actually implement it. So it simply won’t be as nice as theory suggests.