It's a fun comparison, but with the notable difference that that one can compile the Linux kernel and generate code for multiple different architectures, while this one can only compile a small proportion of valid C. It's a great project, but it's not so much a C compiler, as a compiler for a subset of C that allows all programs this compiler can compile to also be compiled by an actual C compiler, but not vice versa.
And it doesn't for the compiler in question either. As long as the headers exist in the places it looks for them. No compiler magically knows where the headers are if you haven't placed them in the right location
stddef.h (et al) should be shipped by the compiler itself, and so it should know where it is. But they rely on gcc for it, hence it doesn't always know where to look. Seems totally fine for a prototype.
So we're down to a missing or unclear description of a dependency in a README - note following the instructions worked for others -, from implications the compiler didn't work.
I mean we know it can be done in little space, given the many tiny C compilers. I think what is most interesting about this one is exactly the creative shortcuts. It's an interesting design space for e.g. bootstrapping to impose extra restrictions.