As far as that goes, the FAA has no control over the certification or
inspection of military aircraft, in any way. They just have to put up
with the fact that the military does what they want with the aircraft
they own.

With one clarification the above matches what I have learned. The
clarification: when military aircraft fly in U.S. airspace, except for
military emergencies related to national defense, they are indeed
subject to FAA flight rules. E.g. hemispheric flight altitude rules, VFR
rules, airspace requirements, and so on. Flight safety would be highly
compromised if there was not a single final authority.

Indeed, they follow FAA requirements to how they use the airspace.

Note in my comments, I was carefull to say that the FAA does not have any
control over _certification or inspection_ of military aircraft.

I guess when I said "the military does what they want with the aircraft they
own" that could be interpreted to how and where they fly them in FAA
airspace, but I was intending to speak to the physical aircraft, not how
they are flown.
--
Jim in NC