Couple things--here's the battery on your mpu, it's the white cylinder soldered in at the bottom-middle:
Screenshot_20190928_061215_org.mozilla.firefox(1).jpg
I'd cut it off asap. It's also possible the battery has leaked on the board to some degree. If you post some more pics of the mpu, we can probably give some further advice. There are several choices when replacing the battery: a coin-style battery w/holder, a memory capacitor, or NVRAM (difference is primarily cost, convenience, and how long they're good for). Without the battery connected, the game won't store certain settings, but that shouldn't matter while you're troubleshooting this problem.
Are you experienced with circuit board soldering?
Another thing: you want to verify proper voltages at the rectifier board. The rectifier board is wired to the transformer, and is typically located in the backbox or mounted on the bottom of the cabinet (you can open the coin door and look inside the cabinet to see if it's there). Those boards tend to have burnt connectors, so you may need to repair/replace that board and the female connectors that plug into it in order to get it sending proper voltages up to the solenoid driver board. There are test points on the rectifier board that make it easy to check with your multimeter.
Once you've verified good voltages at the rectifier board, you'll want to verify good voltages at the solenoid driver board. There are several test points on the board that you can measure them at as well. Looks like you have the original +5v filter capacitor at C23 on that board (the big blue capacitor), it's recommended to replace that, since resets can often be due to low/weak +5v supply to the mpu.
There's also a variety of male and female connectors that should likely be re-flowed and/or replaced on the mpu, solenoid driver board, and rectifier board for long term reliability.
First things first though: gotta verify the voltages. Do you have a meter, and are you familiar with testing ac and dc voltages?