Well, that was interesting. Seems like I don't have a problem at all, I'm just in a very noisy area with zero AM reception.
I can locate and hear the LO go up and down the dial on the other AM radio, and it also has a _very_ loud hum at no station. Also found an area on the dial where I'd receive a loud thup-thup-thup noise on both radios, and heard it echoed when I set the console to that channel and tuned the other one to the LO.
*facepalm* Whelp, time to locate a dial lamp and pop the radio chassis back in the cabinet, I guess
Looks to be a type 51 judging by the "E51" stamped on the base.
You live, you learn. Onwards to the TV part now!