One thing to keep in mind in that video shown earlier (and yes, I know this discussion is a year + old) is that he said he was doing a museum wrap. Museum wraps are on the thinner, 3/4" stretchers, are intended to be framed and the corners aren't seen. It doesn't really matter how pretty the corners are. Aside from the fancy machine he used, that may be part of the reason why he was able to fold the corners so quickly. Gallery wraps, on the other hand, are typically on the wider 1 1/2" (or wider) stretchers, are primarily intended to be hung without a frame, the corners are seen and it's more important then that the corners look nice. My guess would also be that he used mitred corners which are much easier and quicker to do than square corners.
Folding corners isn't that difficult and shouldn't require trimming. It took me a while to get a method down when I started stretching canvases but now that I've got it, it works well. What I do is leave the ends of the canvas at the corners unstapled. Then from the corner, fold the side down at 45 degrees and fold the top flap down flush. There'll be a lower flap created by folding the side in at 45 degrees and this gets folded under and is hidden by the top flap. It is like a hospital corner but that's the way it's supposed to be. You'll see a line of the fold under the top flap but that's fine and is expected. It doesn't create a bunching of extra fabric under the fold. It's clean and neat. With the corner held down I put in 3 staples, one each on the top and side in line with the others and a third at a 45 degree angle across the corner inset a bit from the edge. It gives flush, square, tight corners.