See https://faultlore.com/blah/c-isnt-a-language/ for more detailed analysis.
Also IIRC it was really only UNIX which had this "whatever our C compiler does" mishmash. On most other operating systems it was the other way around, C compilers had to implement whatever calling convention the OS already had defined before there even was a C compiler for that OS (for instance early Windows version used a PASCAL calling conventions, and others (CP/M, DOS, AmigaOS...) some random rules that were most convenient for handwritten assembly code).
PS: and yeah I know that other article (or rather: incoherent rant). It's basically a lot of barking up the wrong tree.
I'd push back on this a bit. The ABI is "put the arguments on the stack this way and jump to this address". The fact that it's easy to do in C doesn't really make a difference, or make that process "C specific" at all, any other language would need to do the same thing.
If you require those arguments to represent rust objects or be reference-counted in some way, it would impose more restrictions on the caller, not fewer.