Perhaps try searching for the specific video title string? Seems a hit and miss solution but trying it just now I discovered someone stole one of my videos and uploaded it to dailymotion! Grrr. I too have tried to figure out the specific url on sites embedding and short of poking around endlessly there does not seem many simple fixes. Wish they simply reported the full URL in YT analytics.
Maybe I'm not understanding the OP's query here. But, in linux - at the command line - you simply point youtube-dl to the URL containing the stream of interest, and it will ferret out the URL for every video on that page _and_ download it for you.