It requires less fuel because air friction works with you during deceleration, not against you, and because, by the time you do the landing, the craft weighs a lot less, not because you only need to slow down.

I think the video left out a lot of the dull part of aero breaking, making it look as if all breaking was done on rocket power.

I do not see how this would be cheaper and safer than just using a disposable run-of-the-mill steel pipe as main engine (yes, that is exaggerating), but they must have done the math.