You should get a proper crimping tool. I'll let others chime in if you haven't got one.
You'll need to replace the two wire connectors (J1 which is 8 pin and J3 which is 20 pin) at the rectifier board. These are 0.156" connectors. We recommend that you use "trifurcon" wire crimp terminals at these connectors which provide greater surface to surface contact. Look at the crimp contacts here and notice the extra fins the trifurcon crimp contacts have:
They come in two sizes for crimping different wire thicknesses. The "18-20AWG" are for the thicker wires and the "22-26AWG" are for the thinner wires.
Please don't forget to buy and install the key plugs that prevent the connectors from plugging in the wrong way or offset.
These example are for 0.156" connectors:
By the way there is a red wire missing from the middle of the J3 connector in your rectifier board picture above.
Also replace the J4 connector at the MPU board. It's a 19 pin 0.1" connector. Because of the battery corrosion on the MPU board, there's usually a good chance that corrosion has reached inside some of the other connectors on that board. Realistically you should replace the crimp terminals on those connectors (re-use the existing plastic connector housings but clean them first). The crimp terminals are cheap.
Replace the rectifier board taking special care to solder the transformer wires (about fourteen) on the rear of the board into the correct "E" positions.
Please feel free to post pictures of how/where you've soldered the wires to the rectifier board and pictures of the connectors you've redone if you want us to double/triple check wires are in the correct spots.
When you're happy, plug J2 only into the rectifier board and power up.
Verify the test point voltages as per post #10 by @frunch above.
If voltages are in spec, switch off, install the solenoid driver board. Plug in the J3 connector into the rectifier board and also the J3 connector into the solenoid driver board.
Make sure the following are disconnected:
J4 at the MPU board,
J4 at the lamp board,
All five displays,
J1 at the sound board.
Power on the game and verify the solenoid driver board test point voltages as per post #10 by @frunch above.
If the voltages all look ok / in spec then power off and plug everything else in.