I believe that's actually what folk are doing these days, literally linking in LLVM and using it to compile C into machine code then executing it during runtime.
That's what I mean by "alternative toolchain," though; some of the code that you are executing at runtime isn't being generated by the normal GCC/Clang toolchain. It's being generated by your custom setup which links in LLVM.
You'd still need to compile your code with a regular toolchain, but you also need additional tools to compile, optimize, and debug code at runtime. That's what I meant by toolchain, even if it is not the conventional (static) definition.