The ever popular Whirlpool design and some others utilize a motor relay that feeds power to the motor when you let go of start switch. If the relay is bad, or has no power, the motor stops immediately. I have seen customers jam something against the button or tape it down to keep it depressed and the dryer in operation.
Whirlpool 27" with Console Open
Whirlpool Even Heat Board
It is easy enough to short out the two bigger wires on a motor relay and see if the motor stays on. A second test is to make sure voltage is on the relay coil when you push the button. The even heat board can also cause this problem. For details on this procedure, view the tech sheet in the dryer console or in my material. The tech sheets for the Whirlpool dryer shown here tell you exactly how to test this motor circuit.
Trick of the Trade: The motor relay and heat relay are interchangeable.