I find it hard to believe their /bin/sh understands . but not source.

source is a bashism. Having a /bin/sh that behaves like a bash is pure luck. Linux had bash as default shell for a long time, but even on Linux, you can't rely on that any more. The *BSDs and commercial Unixes have changed their default shells several times. See https://www.in-ulm.de/~mascheck/various/shells/ for details.

Alexander