good logical work. i'm not sure circuit boards have really helped the operation of simple appliances like dryers. previously relays would have been a little larger and contacts distinct and called out in wiring diagrams and thus it would have made this easier to test, although you were able to observe the damage to the circuit board and remedy. sometimes the damage isn't obviously seen.
if the wiring diagram is clear that the printed circuit delivers line voltage to the circuit that you were lacking but you can't see a reason why, I tend to see if I can manually start the motor by wiring around the circuit board and then replace the circuit board or try testing components if i think i can reason through its design, or send it for remanufacture/refurbishment.