Nice first game. Lets see photos of the circuit boards. Essentially interested if the original battery is still mounted on the MPU. At minimum you will need to be able to resolder the connectors on the board.

Think of your project as a 64 Mustang barn find. It sort of runs but were going to help you fix it up.

First of all, as a former engineer I would absolutely love a project game like this!
You have all the symptoms of a classic Bally in need of an electrical overhaul. I recommend the following on the to do list...

* Rebuild the Power supply board. It needs new filter caps, and you should order replacement fuses for that board. the HV fuse hack is dangerous.
* Rebuild the rectifier board by the transformer. The existing is badly hacked up.
* As for the MPU, very likely the battery has leaked and damaged the board. At minimum you need to remove the battery and replace with an NVRAM chip. However, given the age, I recommend a new Alltech or Weebly replacement MPU.

I would start with the power supply. Recap the filter caps and replace the fuse with an actual fuse. That fuse is an odd duck size but you can buy replacements. They are available. Check the HV and try to adjust it to around 185v or so. Resolder the connector pins and the flipper relay connections.
From there replace the rectifier board with a new style and rebuild the connectors.
Along the way make sure the game boots and the voltages are correct. When you finish rebuilding the PS and Rectifier and the game still boots then rebuild or replace the MPU.

I've seen the posts about modding Bally boards to be 'bulletproof'. I'm reluctant to do so only because it does involve hacking up the boards and opens up the possibility of introducing another fault. There are thousands of unmodified board sets out there working fine with a few minor updates. If you do want to perform the mods I would recommend sending the boards out to a pro unless you are very confident on your tech and board soldering abilities.