Just spitballing here. Have you checked that the contacts arent touching. I switched out rubbers once and the new ones forced the contacts to be closed. Also, the next thing I'd do is check for continuity to the pins and then continuity on the board and any solder issues or wires inadvertantly touching. I would also reflow the solder on the board connections. Have you checked the fuse holders?

You said you switched some transistors. Did they test bad before you changed them?

Anyway. Just thinking of things you haven't mentioned. Good luck.

1. Why wouldn't it be the wiring from the board to the solenoid? Have you checked the continuity from the board plug to the solenoid plug?
2. Not sure if the fuse holders would have anything to do with it but they are notorious on Data East machines. check proper fuses for continuity and press down on the fuses to check and see if they're cracked/broken.
3. reflow all associated pins on the board.

Have you switched out the board to another DE machine? Reflowed the pins?