Ordered an untested Ms. Pac-Man board off eBay for a fair price.  As an aside, remember that “untested” means the exact same thing as broken, especially when dealing with older hardware.  Older hardware (games) should be presumed to be broken unless proven otherwise, so bid/price accordingly.

Its Z80 Sync Board was plugged in backwards, but surprisingly it did not get smoked and still worked!  The game had lots of trouble coming up (sometimes static garbage, sometimes partially into the self test) but at least that verified that basic functionality (CPU, RAM, ROM) was working.

Started pressing on socketed anyone and discovered that the behavior changed when I slightly clocked the Z80 Sync Board in its socket.

Pulled it, removed and replaced its old socket, and the game came up correctly. Easy fix. The only tricky part is being careful not to lift traces or the thru-hole plating off of old boards like that.

Installed a ROM with the 1P/2P cheat hack where holding down 1P makes you invulnerable, and holding down P2 gives you a speed boost, giving you the option of playing normally if you choose or cheating your way to victory!