This has already been discussed on email-devel but didn't come to a
conclusion.  I don't want to linger on it, but I would like the problem
to get assigned a bug number, so it has a chance of being addressed in
the future.

A summary of the problem: suppose buffer A has a buffer-local value of
'process-environment' containing an extra directory in the PATH, and
that a given 'program' is only available on that directory.  Then 'M-x
compile RET program RET' will not find 'program', because the
*compilation* buffer will not "inherit" the PATH from A.

For the reasons why one would want/need such a setup, see
https://lists.gnu.org/archive/html/emacs-devel/2021-08/msg01380.html
and other messages in that thread.

Finally, I attach my proposed patch.