Would this cheat puff pastry really work?
Yes, that will work just like traditional methods. That's the real deal, it's not even a cheat, it's just smart.

BTW, the last line in your question raised my eyebrows. In puff pastry there is no waiting "for the butter to get softer so that it flattens". The butter is flattened (or sliced) with brute strength while the butter is still ice cold. That's critical to puff pastry. If the butter melts even a little before the layers are formed and it goes in the oven, you're sunk. The final product will be greasy and won't puff as much as it should. You get that puff by having very distinct layers of butter between layers of dough. If the butter is allowed to get warm, even a little, it will absorb into the flour.

Paul Hollywood's method is brilliant. It almost makes me want to make puff pastry from scratch. (almost)

For those disinclined to watch the video but still interested in puff pastry: What Mr Hollywood did in the video was grate frozen butter, then he kept the grated butter in the freezer until he had the rectangle of dough ready for the butter. Then he just sprinkled on the frozen butter and did the first envelope fold. He rolls that out into rectangle again, adds more frozen butter, and repeats the fold. Fridge for one hour, one more roll, fold, roll, done. Well! I wish I had thought of that.

EDIT 5 months later:

I'm still not particularly interested in making puff pastry from scratch since I doubt I can improve on the frozen stuff, however, I was encouraged by @rumtscho to try my hand at croissants. Since croissants are basically puff pastry with yeast, I decided to form my croissant dough using Mr Hollywood's technique.

It worked like a charm.