I don't recognize the ref, and haven't heard through the ref grapevine about the incident in question, so I can't speak specifically to it, or what happened afterwards (other than from the video, it seems obvious that the ref blew the call).
To some of the other points you raise:
1. "It would be a shame if the ref couldn't swallow his pride and admit his mistake."
- that's a little more nuanced than might think. Every single tournament I ref, there are coaches that tell me I don't know the rules, or tell me to get the calls correct, etc. 99.99999% of the time, they are the ones that are mistaken, and are simply speaking from emotion. So you tend to disregard a lot of what coaches say. I don't think I've ever changed a call based on something that a coach said to me. But then again, I'm 100% certain that I've never blown a call like was shown in the linked video. Refs do replay sequences in their mind; that's part of the learning process, and how you get better. Most of the changes I've made are things like where the table totally screws up the score, particularly in youth where you can have a TD, then 3 NF, then a reversal, then 2 NF, then an esc, then a TD, then another reversal, etc. in a 45 sec. period. The coach will question the score, so then you have to go to the table and work through the entire sequence of events from memory, and correct whatever mistakes they've made.
2. "Mistakes happen. I've never reffed, but I've often thought that a ref must start getting almost dizzy after reffing 100s of matches in a short period of time."
- I wouldn't call it dizzying, but I'm in very good shape. In tournaments, you get mental fatigue. I've done things like raised a kids hand as the winner after the 2nd period, or forgotten who had choice in the 3rd period when I had just flipped for it a minute ago, etc. Those are things that you can easily correct, and I always apologize and indicate that it's my mistake. Come to think of it, I recall one incident which can be used to answer both of these points. It happened in a day-long tournament, near the end of the day, and I was mentally exhausted. In the 2nd period, I totally forgot who started on top/started on bottom. I think there were a couple of stoppages for OOB or for stalemates (or some combo). If I recall correctly, the kid on bottom kept getting almost out or around, but never enough to award an esc or a reversal. Then late in the period, after some flurry, he ended up getting a reversal, but I never signaled the points, as I forgot who started where. The coach challenged what had happened, and after checking with the table that COLOR had started on top, and OTHER COLOR ended up on top, I did award a reversal.
Back to the linked video -- it all depends on how the tournament was set up. Sometimes we're lucky, and there are 2 refs on a mat, and you set up some type of sequence (say ref 1 does 2 matches, then ref 2 does 2 matches, or other variations), so there is another ref that might have been watching the match, and could after the mistaken call for a fall tell the other ref that it wasn't a pin, and the ref that called the pin could correct the call. However, there was a shortage of refs before the pandemic started, and the pandemic has made the situation much worse, as a large number of older refs have retired, and others that are immuno-compromised have elected to go inactive. This season, I've done a lot of tournaments, but only one of them was set up so that two refs were at each mat. In situations where there is only 1 ref on a mat, the only person that can change the call is the ref that just made the inaccurate call. Perhaps he could mentally playback the sequence, and realize that he made a mistake and correct it. But he's the one that just made the mistake, so in his memory the kid may well have been on his back and not on his stomach.