During my only private session with a certified barbell instructor, he taught me to do this as well. I didn't get it right (injury made it hard to breath and move hips) but he only told me to do it to clear the bar path, because otherwise I tended to do a rainbow motion around my head with the bar as well as move my NECK back to avoid the bar, instead of actually tilting my pelvis forward while tilting my chest back to clear the path.
I don't think it's supposed to add momentum to the bar.
DISCLAIMER: I'm only guessing here, so correct me if I'm wrong:
In the video, the first guy cheated his first rep which gave the bar momentum. His other rep was OK.
The lady's reps in particular seemed best. She would tilt backward and launch the bar - I don't feel her tilting around added momentum at all.
The first guy's second set IMO is more correct. He didn't "cheat" his reps.
So aside from the first's guy's first rep, and very few reps throughout the video, I don't think the bar launched like a rocket due to anything but their normal pressing strength.
Yes, I did the breathing at the top and a stretch reflex, or rather, I didn't breath at all However, I didn't really cock my hips back properly, I would usually just move my neck out of the way, so that's something to correct. Timing should just be to move out of the way at the last instant possible, I think.
I can tell you however that the stretch reflex WORKS and prevents the bar from stalling midway through the rep forcing you to grind through the rest.
I discovered that the few times my workouts looked like this:
4 reps, 5 reps, 5 reps
... were when I would pause at the bottom during the first set, and then use the stretch reflex and complete my sets a bit quicker in the second and third sets.