all you can really do is stick a voltmeter on the handle release coil and verify you have constant voltage across the coil lugs.
if you do, then trace the controlled wire on the coil back to where it goes ... maybe the cpu board in your case, but you don't have typical power transistors like a TIP 102 that use a TO-220 package on that board ... then follow the board trace from the connector to the transistor that drives the coil. Traditionally on bally games the common power wire on the coils is orange, so the controlled wire is the other one. No guarantee J&T did it that way tho.
if you have a logic probe and can tell the part number of the transistor (so you can identify the legs if the board isn't marked), you can see if the gate of the transistor is turning on/off. A voltmeter would work too (and if you poke the wrong leg it won't blow up like a logic probe might).
you can also test the transistor to see if the it's shorted (how depends on what the transistor is) ... just google transistor testing for instructions.
without docs you're in reverse-engineer mode. A typical circuit is a power transistor drives the coil and a decoder/buffer/opto isolator chip drives the power transistor - possibly thru another transistor. The cpu drives the decoder/buffer/opto chip. The idea is to prevent damage to the cpu if the power transistor fails.
if you are still stuck and want to email high res pics to slotpics@cdyn.com, I'll give ya some bad ideas on what to try. Mostly need to see the part numbers and the traces on both sides of the board.
was there a transistor in the obvious spot or do the solder pads look unused?