Try removing the short ribbon cable between the CPU and driver board and seeing if the CPU board boots up.

Check ALL the fuses and verify they are all the correct values.

You need to see if something is drawing down the power supply when it energizes (ribbon cable) or is over fused.

If that checks out, check the 5V regulator circuit. Start with measuring the input to the 5v regulator section (across C203) which should stay over 10V

If you are sure the 5V is stable, it sounds like the CPU is not running reliably. If it locks up, the blanking will engage.

Do you have a logic probe? If so, when it isn't running correctly, try probing the pins of the CPU to see if it is running. Start with pin 35 (clock) then the address lines (8-23) and confirm they are oscillating. Also confirm 5V is reliably, locked on at 5V all the time.

Is the CPU board original to Southpark or did you get it from another game?

From the schematics:
A watchdog is used to monitor the CPU and the 5v supply. If the 5v supply is below 4.75 the watchdog will hold the CPU/Sound Board & I/O Board in reset. The watchdog must be fed at a rate of 250ms or faster. The signal used to feed the watchdog comes from the EPROM Bank select signal used to load U211. The CPU has a timer interrupt used as a heartbeat for the system this signal comes from counter U2. The clock for this counter is the CPU Q CLOCK. Clearing the timer interrupt is done by reading the DIP Switch. The timer interrupt can be observed at TEST POINT FIRQ. In normal operation "FIRQ" should be toggling at a rate of 976Hz.