Unfortunately this is not possible to do in a generic, robust, and stable way. Youtube changes its website design extremely frequently, so there's not a good way to fetch the information from the page. And adding this functionality for Youtube would be the start of the rabbit hole for other video websites.
My suggestion is to add in "Anki card format" an option where Yomichan copies the embed link of a video.
Like this:
This would make it easier to mine sentences from any YouTube video.
Unfortunately this is not possible to do in a generic, robust, and stable way. Youtube changes its website design extremely frequently, so there's not a good way to fetch the information from the page. And adding this functionality for Youtube would be the start of the rabbit hole for other video websites.
There is something called Mpvacious, by the way.