I have a friend who has the abovementioned RCA tv, and would like to get it going. Alas, no remote.

Most everything in the ultrasonic remote days would work without the remote. So it could be fixed and used as a non-remote set while waiting for the remote to show up...if everything is serviced properly the set will either work fine with the remote when it shows up, or only need a few adjustments of the remote receiver chassis.
Also there weren't that many sound frequencies used so a common 4 function Zenith remote or a higher button count ultrasonic should work some or all of the functions.

The RCA number for the transmitter is CRK10A, part number 10T111. It's an eight button ultrasonic unit that was also used with the CTC28 chassis. They show up on E-Bay quite often. Make sure your friend has a remote set. The CTC30 has power tuning, but not all chassis were equipped for remote control. The remote set model numbers all ended with "R", and the on-off on the volume control was turn on-off rather than push-pull on-off for the non-remote sets.

Has the channel change chrome bar on top. Not sure about the remote now