Legality of a flight

All true, but there is no specific "Angel Flight" exemption.

Sorry, didn't mean to imply that there was a special exemption. Angel
Flight is specially recognized by the FAA. If you tried to start the
same organization today the FAA would require a 135 certificate since
it fails the "common cause" test for 135 operation. The FAA has
threatened to shut down Angel Flight in the past but they do a good job
of showing successful missions on "Good Morning America" and the like.
The FAA chooses to ignore Angel Flight.

Our flights are purely part91 flights with no "compensation" going to
either the pilot or the organization in return for the flight (neither
the patient nor the hospital is ever billed a dime).

True, it passes the compensation test but if failes the "common cause"
test. Anytime you publicly offer to transport someone from one place to
another that you would not have otherwise gone it, its generally
considered part 135. AOPA has a good publication on the several tests
the FAA uses to determine part 135.

-Robert