I tried Reece's suggestion and wiggled two tubes in their sockets. Turned the radio back on, and it works. Sounds wonderful (like all Zenith radios of that era, late '50s-early 1960s), and brings in more stations than I can shake a stick at, just using the built-in line-cord antenna. (I often wonder how many FM stations this radio could receive if it were connected to an outdoor FM antenna.
) I live near Lake Erie (within one mile of the south shore) and can get stations on this thing from Cleveland, Akron (a small city 30 miles south of Cleveland), Ashtabula (a city on the Lake Erie shoreline some 50 miles from Cleveland), and quite possibly several out-of-town stations as well.
The 12AT7, 6BJ6 and probably, even likely, most of the other tubes are more than likely quite weak, as most of them look as if they are the originals from when the set was new 53 years ago. The radio likely quit, replacing all stations with white noise, because the 12AT7 was not doing its job; however, the tube itself, other than being weak, is probably OK since the radio played perfectly after I reseated the tube in its socket. And I mean the set sounded great! That great Zenith sound, no discernible hum -- I bet this thing sounds as good now as it probably did when it was new. The dual-speaker arrangement with a tweeter as well as the main speaker is a hallmark of Zenith, as they used this arrangement in many of their higher-end radios as well.
Lower than normal B+ voltages, as VK member dieseljeep suggested, may well be due to a weak selenium B+ rectifier. After all, if the one in the radio is the original selenium (I think it probably is in mine), it is over five decades old and could short or open at any time. I will be on the lookout for the telltale rotten-egg smell and/or smoke of a shorted selenium, if or when, but since the radio seems to be working very well at this time, I'm not going to worry about it. However, if it does short, open or otherwise become defective, I'll replace it with a 1N4007 diode and a dropping resistor in series with it, as is usually done when a selenium stack goes bad for any reason.