When I select Listen/Phono, I get a loud rumble through
the speakers, whether the TT is on or not, and whether
it is plugged into the back of the REC or not.

I don't know what you mean by "rumble", but I assume you mean "hum".

Yes, if the TT is not properly grounded to the amplifier, there's a good
chance of getting loud and annoying hum.

On the other hand, the fact you get this noise even when the TT is not
connected suggests there might be something wrong with the amplifier's phono
preamp.

I'd suggest starting your troubleshooting by making sure the TT is properly
grounded to the amp. Check the ground wire to be sure it isn't broken.