SUCCESS!
While looking for a relay among my collection of such devices, I also thought that the problem would likely not be the relay. I have used hundreds of these over many years for work and home projects, mostly 12V and 24V coils, AC and DC, and have never had a coil fail.
With that in mind, and before going any further, right after lunch yesterday, I removed the capacitor and tested the relay with the 9V battery. YES - I heard it click! So the relay is OK; well, at least the coil, that is.
The question now was the 7809. I remembered that Stan said, "If the 7809 is indeed seeing a shorted relay coil, it will get hot real quickly and go into thermal shutdown to protect itself." Now we know that it did not see a shorted coil, but a bad capacitor instead; but still, it should have "...shutdown to protect itself", so I soldered the 7809 back on.
Next was finding a replacement capacitor for the 220uf 16V failed cap. I have dozens of small electrolytics, but only one 220uf, and it was 25V, so slightly larger in physical size. It fit in the space, but it blocks one of the two holes for the screws that hold the PC board, but I did not deem to be a critical issue, so I soldered the cap in place.
And everything works fine now!
(I was concerned about putting everything back together, as I have it apart for several weeks, but the reefer went back together nicely).
Thank You, STAN, and everyone else who chimed in.
Alex