I decided to touch up all my soldering on the chip sockets. Must have had a bad connection on one of them, because now I get a working display.
Checking continuity is not a reliable way to test solder joints. Many intermittent joints become conductive when you poke them with a meter probe. Visual inspection is more reliable. If it doesn't look smooth and shiny, it probably isn't right.
Yeah. Although, for the record, I only tested the tube connections that way, and in this case the test was valid because I wasn't poking any solder joints with the probe; I was testing from the tube leads (where they exit the glass) to the output pins of the header, not touching any solder points. So they seemed okay, and that led me to the chip sockets, and although they were all shiny, a few of them were a little scanty, so I touched them all up and that solved the issue.
I figured the flashing was just power-off alert or set-time alert or something. Once the switches were installed and I set it up it was fine.
Thanks for the reply!