Most times they actually are a normal .c/.h combo, but the implementation was moved to the "header" file and is simply only exposed by defining some macro. When it is actually a like a single file, that can be included multiple times, there is still code in it, so it is only a header file in name.
What is the big deal in actually using the convention like it is intended to and name the file containing the code *.c ? If is intended to only be included this can be still done.
> avoiding the C standard library are regressive and obsolete
I don't understand this as well, since the one half of libc are syscall wrappers and the other half are primitives which the compiler will use to replace your hand-rolled versions anyway. But this is not harming anyone and picking a good "core" library will probably make your code more consistent and readable.