The problem was the enhanced tracking protection of the Firefox, which was shown by a blue badge to the just beginning of the url bar, something like this:
This was discovered by me from this meta solution. As evident in this meta solution, this problem would not only stop YT videos, but also twitter links/videos etc. And may be some other external media links too. So all problems are solved now.
As soon as I turned off that protection (I’ll search for as to how can I turn off that nuisance protection for all sites, forever, in one click).