> plus the fact that the kernel doesn't know anything about PATH (it's only your shell that does)
glibc is what deals with PATH, not the shell. On Linux, it's execvp(3) in libc which is implemented in terms of execve(2) in the kernel, but POSIX doesn't make a distinction between syscalls and library functions, so a kernel could implement either or none directly.
To be fair, many shells have command caching, so they effectively roll their own exec syscall wrapper: