When you ground Q58 to make the motor run, you are actually "bypassing the transistor" and turning the motor on manually. This proves that from after the transistor in the circuit, that everything from the main board connector, the wiring, the relay board and then the motor all are good. This does indicate that in test or game mode that Q58 is not turning on, or the circuitry just behind the transistor is bad. Soooo, your easiest and first step is to REPLACE Q58. It is the most likely failure. Take out your book and look at the circuit for Q58, you can see a smaller transistor that turns it on, then the signal comes from an IC before that. To be sure, replace both transistors and this will 99% sure fix it without much other troubleshooting or hunting around the board.
Just be sure and pay close attention or write down the wire connectors when you remove them to take the board out. All the connectors are keyed so as to go in only one way BUT not so much. Use your digital camera and take some pics of the connectors to make reassembly go much easier for you.
Hope this helps...