They are certainly bad vendor toolchain, but I want to push back against the idea that this is a general C problem. But even for the worst toolchains I have seen, dropping in a pair of .c/.h would not have been difficult. So it is still difficult to see how a header-only library makes a lot of sense.
One of the worst I've experienced had a bug where adding too many files would cause intermittent errors. The people affected resorted to header-izing things. Was an off-by-one in how it was constructing arguments to subshells, causing characters to occasionally drop.
But, more commonly I've seen that it's just easier to not need to add C files at all. Add a single include path and you can avoid the annoyances of vendoring dependencies, tracking upstream updates, handling separate linkage, object files, output paths, ABIs, and all the rest. Something like Cargo does all of this for you, which is why people prefer it to calling rustc directly.