Yes, is possible, and not uncommon. You may wish to dock it to prevent excessive rising.
Some pastries are made with puff pastry that is already baked separately, such as the famous Napolean or Mille-Feuille . For example, in this recipe (which uses commercial puff pastry), the instructions are:
Preheat oven to 425°F/220°C. Line 2 baking sheets
with parchment paper or plain brown paper; set aside.
Unfold puff pastry sheet and trim edges to a 9-inch square. Cut pastry
into nine 3-inch squares. Transfer pastry squares to the prepared
baking sheets; prick pastry. Bake in a preheated oven for 18-23
minutes, or until golden. (Or bake according to package directions.)
Carefully remove pastries from baking sheet. Cool on a rack.
I cannot endorse the plain brown paper idea, but the method is fairly standard.
Here is another example, from King Arthur flour, of croissants du patisserie, which is essentially puff pastry baked as a croissant (as opposed to croissants du boulanger, the baker's croissants, which are yeast raised as well as a laminated pastry).
Puff pastry is a laminated dough, with very strong gluten development, so an extra couple of days in the refrigerator should not have caused problems.
450 F seems like a typical temperature, and the time seems in the normal range.
The only thing you have mentioned is that is definitely outside the standard treatment is trimming the edges with a butter knife. Normally, you want to trim or cut puff pastry with a very sharp knife or pizza cutter, in order to cleanly cut through the layers. A blunt knife like a butter knife can mash the layers together, making it hard for them to separate at the edges of the pastry.
Still, this should have lead to lopsided or strangely risen pastry, rather than a complete failure to rise, especially in the center.
The other possibility is that the dough was too warm when you rolled and worked it, or that you rolled it too much, which would work the fat or butter layers into the dough phase, rather than keeping discrete layers of flour then fat, which is what allows the rise.
Best Answer
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.