I have an early 2015 MBP with Big Sur, and I was having the problem described above not only for videos played within Safari, but also for MP4 files played with Quicktime Player. I found another solution online that worked: Clear all cookies in Safari (Preferences > Privacy > Manage Website Data > Remove All) and then reboot. Why that fixed the problem for Quicktime Player as well as for Safari, I don't know. The other strange thing is that all the cookies I deleted are back, but videos are playing fine.