No, not kidding. Surely the FAA/FCC would let someone set one up
in Arizona or something

The FAA couldn't care less. If you don't jump through the hoops to
have it inspected, you can still have it shown on the sectional but it
will be noted as VFR use only.

The FCC will require you to license the transmitter, but it's very
doable.

There are NDB's sitting on private fields. Some even have private
approaches charted, others are VFR only.

Michael