Seems more to me like an inexperienced driver or someone who went into a blind panic and couldn't think straight and made up a story to cover their ass. Of course the foot brakes would work. Unless he was trying to brake lightly over a long distance which would have boiled the brake fluid and induced fade rather than just stamp on the brakes, switch the engine off and stop the thing. I bet the fire was just the paint from the brake pads burning. I had a couple of cars that would set the pads on fire if you were pushing hard and late braking too much.
In most autos, older ones anyway, dropping into P slots a pawl into a slot in a cog which locks the transmission solid. Can't speak for new cars, I outsource all my transmission work. When moving it doesn't work because the cog is spinning too fast for the pawl to slip into the hole and it makes a horrible girrrrrr noise. Then all of a sudden at around 2-3 kph the car will jerk to a halt.
Switching the engine off is a good idea. If the engine is stopped, the car can't keep moving. The only thing to be aware of is that you don't activate the steering lock. Lose your steering and there's a good chance you will hit the barrier and flip onto your lid.