Kinetic energy is relative too. Two cars moving at 100MPH in the same direction (relative to the reference frame) have no kinetic energy relative to each other.

You're right that if you match your reference frame to one car, that car becomes "stationary" until the collision, but after the collision it is moving relative to the reference frame. But that's the key--it now has kinetic energy relative to the reference frame, but not to the other car. As far as the pair of cars is concerned, all of the kinetic energy between them has been expended, because they are now moving together.

Of course this is assuming a scenario where the cars crumple into a single mass, i.e. they don't "bounce". I think part of the confusion in these comments stems from the problem being poorly defined. Some people are considering crumple zones, others are considering whether a brick wall is movable and still others replace it with another car for simplicity.

I do think the driver's ed manual is correct in some Spherical Cow sense, but I don't think its authors ever intended for the problem to stand up to Hacker News scrutiny. It's quite funny to me how innocuous things like this can generate these "unladen swallow"-type discussions on here, even though I'm participating.