If the loud noise is kind of a hum coming from the speaker you may have a 12v problem. Usually its the isolated ground return for 12v on the driver board. When that connection fails the 12v filter cap on the driver board is no longer doing its job and a nasty hum comes out the speaker. Using the test points on the driver board or just check across the big C23 filter caps leads for at least 12v(reads more like 15-18v usually). If the 12v is really low check the 12v on the rectifier board. If its good on the rectifier board but nothign on the driver board its probably the orange color 12v return is burned up at the driver board end.

Also consider doing the driver board ground and 5v mods as shown on the guides. If there is a problem with that 12v return orange wire between the rectifier and driver board, the driver board ground mods would band aid that problem.

probably need to replace both the male and female side of the connectors on the rectifier board, specially if they are burned up. Best guess is the thinner solid orange wire on the 20pin rectifier board plug is the cause. When that orange wire goes out of circuit the sound board will start to hum.
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check out the ground mods on the driver board. they help band aid bad connector issues by relieving stress on some connections that like to burn.

https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#Solenoid_Driver_Upgrades

Also, install a fuse on the SDB instead of the aluminum foil. May not be related, but that's kind of scary.

The color codes for each wire is in the full manual w/ schematics up on IPDB (if you don't already have a copy).
Bottom middle of the "Plug Connectors" page has the Wire Color Code key.
So example, J1, pin 1 shows as color "50", which translates to 'white, no trace'.
Or J3, pin 5 - color 92 = 'grey with blue trace'.

The transformer/rectifier board is the easiest to do first, because its just a matter of testing the voltages at the TP's, and then resoldering/re-pinning the connectors.
From your pictures, my main concern is the battery damage on that MPU board.
Most repair information can be found on the https://www.pinwiki.com/ early Bally section. (including how to build a bench tester, although I've personally never done it... I have several Ballys, so i tend to just swap boards into working machines to test.)