Expected behavior for damaged transistor
Remove R4 and confirm that the motor remains off. Shorting R5 is not sufficient to be sure, there could be D-G leakage on Q4 for example. I would suspect Q4 first, damage to Q3 would likely be more of a random failure (toys are not known for using the finest space-qualified components).
Semiconductors do tend to fail &quot;on&quot; unless they've been abused to the point where the case is blown apart or wirebonds are vaporized.
It would be better if there was a diode across the motor unless there is some reason that is not shown in this schematic that would make that a terrible idea (possibly damaging other components).