rustc is only loosely tied to LLVM. Other code generation backends exist in various states of production-readiness. There are also two other compilers, mrustc and GCC-rs.

mrustc is a bootstrap Rust compiler that doesn't implement a borrow checker but can compile valid programs, so it's similar to to your proposed subset. Rust minus verification is still a very large and complex language though, just like C++ is large and complex.

A core language that's as simple to implement as C would have to be very different and many people (I suspect most) would like it less than the Rust that exists.