There's not much left that would make the entire display go blank intermittently.
Try a light tap here and there to see if it's a bad solder joint. The IC's are sensitive to static electricity so don't go for a walk and then touch the PCB.
I would ensure the MM5314 clock IC's /output enable pin 1 is grounded (i.e. connected to pin 2 VDD=GND). If that pin floated then it could cut in and out. You can check with an ohmmeter. U2 should also have power on its pin 1 +15VDC, or ohmmeter pin 1 to the new caps. Old IC sockets can oxidize too.
Note the Jameco JE725 schematic is missing pins 1,2 and 10,11 (12/24hr, 50/60Hz) jumper.
If the clock IC's multiplex oscillator stops (C2/R9), the display usually stays stuck with only one digit lit, it stops scanning but doesn't go completely dark as I recall. I'm not 100% sure though as the IC has a 100Hz min. mux frequency, like it's synchronous.
The clock IC's max is 20VDC and U2 CD4049 is 18VDC, so a power line surge might have happened which can damage an IC. It's kind of a last guess. Some 12VAC adapters have high output, especially at lowest load 1:11:11 for fewest LEDs lit compared to 20:08:08 for max. current drain.
MM5314 can still be bought from scalpers, but it is expensive.
edit: I just tried stopping the mux oscillator by grounding pin 23 through a 10k. The display stops scanning and lights up only one digit, but the time keeps advancing. Could not make the display go blank that way.