He doesn't say why. He just says that it happens. You can see the brake fail alert on the dashboard.
In the video, when he hits the brakes the car slows down sharply, even with the gas on the floor, but he lets off the brake for his “don't pump” part.
Obviously, the video has been edited. You don't see him pumping, but you see the brakes in a failure mode as a result. You have to let off on the brakes when you pump, that's what pumping means. You see the car slowing somewhat before the brakes fail, and he says that's not enough to stop the car. He claims that once the brakes go into their failure mode, they are totally ineffective.
Unless your minivan is a Toyota, your experiment might be meaningless.
There may very well be two problems that augment each other. There's the initial acceleration problem, and then there's the brake failure. The first may well be mechanical, the second is almost certainly software.