In my MX-5 a normal push of the button only stops the engine if the car is at a standstill. To kill it while rolling, you need to do a long press (according to the manual). Of course, a stuck throttle pedal can be mitigated by pressing the clutch and/or putting the gearbox into neutral... which might perhaps not be possible with a modern, fully electronically controlled automatic gearbox.
By the way, the manual for the MX-5 has 15 pages (!) on starting and stopping the engine, not counting the chapter on the stop/start system. The whole manual has 603 pages. No, I haven't read it.
Anyway, I'm sceptical about the story. Most "unintended accelerations" turn out to be the fault of the driver after all...