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random sample: https://youtube.com/watch?v=D0tI7TMYMJ0
tried "embed" trick and another bypass age restriction service and I may found the reason:
"Playback on other websites has been disabled by the video owner."
After much testing and trial and error, the best I can get is what you mentioned about the video being disabled on other websites by the video owner, yes this is because the best attempt is to get the video embedded into the YouTube page in an iframe instead of having the content directly loaded by YouTube's servers. This makes the video think it is being loaded into a different website even if it is just an iframe in YouTube's website. Unfortunately I do not personally know a way to make it see that it is being embedded into a YouTube page (which should make it fine haha).
For now, I have no solution. I thank you for reporting this!