I understand the reluctance to pop off the plate, you might be able to reattach it with small screws.
Resetting the brushes can be a little tricky, but the housing for the brushes is typically held in place by 2 nuts, inside, on the same long bolts running through the whole motor. You'll have to disconnect the motor wires from the plug, but the wiring looks good as does the grommet on the motor.
Once you have it open you can post more pictures if you need any help.