Ni-Cd battery is not working properly
Try putting the car up on a block so that the wheels don't touch the ground and operating it from a bench supply. Note any problems and also note the peak current draw.

NiCad batteries in general are known for having very high peak current capacity (at one time, large NiCad batteries were used to start jet engines), but as they get older, this parameter is one of the first to fail, and maybe your battery is no longer capable of supplying the starting current required by your car's drive motor. Try measuring its terminal voltage with various resistive loads, up to the current noted in the previous paragraph.