I'm curious: Is there a programming language or a compiler that is actually smart enough to make that distinction? In principle, C++ compilers wouldn't have to monomorphize template instantiations all the time: it's in principle possible to compile templates in a way that adds a hidden "type" parameter to the function which essentially acts as an out-of-band vtable to provide the operations that the template uses (yes, there are ABI issues).
For example, due to operator overloading, given "T a, b;", you don't know whether "a + b" should be a simple addition or a function call. For full genericity, you'd want to compile this as a vtable'd addition, but that's going to severely harm performance for primitive types. So you'll have to monomorphize for primitive types. This applies to every operator in C++, of which there are a lot.
But that's just performance. Worse, the semantics of operators can depend on whether they are overloaded or not. For example, "a && b" short-circuits normally, but does not short-circuit if "operator&&" is overloaded. So now you need a variant for whether or not operator&& is overloaded (similar for "operator||", and even "operator,").
I'm fairly sure I've only scratched the surface of weird things that come up with compiling C++ templates. There's loads of other fun examples of template weirdness, like for example the program that throws a syntax error if and only if an integer template parameter is composite: https://stackoverflow.com/a/14589567/1204143
Monomorphization would ideally be treated the same way.
Yes, there are aspects of C++ that make that so difficult that it's just not realistic, but really I was only bringing up C++ because the parent poster certainly meant the C++ std::sort. The same argument could also be applied to other languages that have less baggage, e.g. Rust comes to mind.
If this question intress you, it is worth watching.
If the comparisons are expensive enough to warrant not inlining them, then I very strongly doubt you'll find a meaningful speed difference between qsort and std::sort. I don't know if you're concerned about code size and instruction cache etc., but I doubt that matters if comparisons are expensive.