Count the LED-light flashes on the board .

Also look for corrosion from a leaked battery

https://pinwiki.com/wiki/index.php/Bally/Stern#MPU_boot_issues

Battery damage in the power on reset area can lead to this symptom. If the reset signal floats or is stuck always high it sometimes just works but you need a few power cycles to get the CPU running ok. Bad 12v filter cap too.

When you turn on the game you should see a brief low signal at CPU pin 40 then it goes high and stays high. A logic probe will show the low to high transition if reset is working okay. If you don't see low to high at power on, there is a problem.

Measure ripple on the 12v filter cap. DMM set to low range AC volts and go across the 12v cap leads, much more than 0.25vac is generally considered too much for this system.

Does the board have a memory cap or battery backup for the 5101. I had a some boards that would not boot reliably until I put the memory cap in. Are you getting any flashes when it fails to boot? I had a no-damage board that had a loose IC socket that would sometimes hang on the flash for that IC. Reseating the chips will sometimes corrected the issue.