In the scene where Doc Brown is trying to hook up the cable to the tower after a tree crashed on it, Marty is back trying to get the DeLorean started. Look closely..when the scene shifts to his foot, you will noticed that the car is a stick shift. Any car that is manual transmission, will not start unless the clutch is depressed. The car was not doing anything!
Not all cars have a clutch safety switch!! And sorry to say that the De Loreans also did not come with that feature.... They were made over seas and it was not mandatory... sorry........
Here in Britain nearly all cars are manual. I drive one myself and can therefore confirm that if the car is in neutral, it will start with the clutch up. If the car is in gear and the clutch is up, it will jump forward (or backwards if in reverse) and stall.