BitTorrent may have been big as in number of files, but as far as users and having content on demand it never got there. I remember waiting for days to get a single movie, not because my Internet was slow, but because the peers were slow.
When it comes to a YouTube replacement I don’t think you are going to get big relying on users to be the servers. Nevermind the fact that the nature of how BitTorrent works means no company will allow their content on it legally.