Look in the first 5 seconds or so. That is one long belt. However, I still agree with you. He shot those nails for a long, long time. A person would have to watch almost the entire video very closely to see if they could find a break where they loaded a new belt.
Back in my machine gunning days, the assistant gunner could clip on a new belt fairly easily for continuous action without stoppage. I don't know how hard this is to do with a nail gun.