Which musl issues are to do with GCC? Alternate C libraries are common on Linux e.g. uclibc, dietlibc, bionic... Not to mention also the other OSs GCC runs on that don't use glibc. Of course, mixing C libraries between the main executable and libraries probably won't work.
As there are common people on forums searching for problems with their C code crashing and burning, because while those libraries conform to ISO C standard, their implementation defined semantics aren't the same.
But the original post's meaning of "independent" runtime library was "compiler independent" (vs C++). Not that there was no difference between the C libraries.
This makes no sense. As I said the original poster was saying that g++ and libstdc++ are tangled together and it is not possible to use g++ with another implementation of the C++ libraries. But you can use gcc with other C libraries. As proven by gcc running on other OSs with non glibc libraries.
If anything isn't "independent" here (using your definition) it's the apps that rely on implementation detail - not the C library, the C programming language, or the compiler.