I had incorrectly assumed tramp-own-remote-path would prepend the remote
system's PATH to tramp's exec-path, but it does not, it prepends your
local machine's PATH onto it. tramp-default-remote-path was what I was
looking for and it is already in tramp-remote-path by default.
If tramp can't find a program on the remote it's because either the
remote doesn't support getconf (which tramp uses to scrape the remote
PATH), or your PATH on the remote has been incorrectly configured.