Ahh...Should have read my own link... They are quite oldschool indeed. Re-branding and restricting the service to games does indeed make sense coming from those days, and Google kind of has an unfair advantage by having so many eyes on it that they can enforce the rules necessary for allowing people to live stream whatever they want.
It's interesting to think about how many different ways video streaming platforms have been shaped into different audiences (private/public, contacts/strangers, content based or not, self[users/mods]/central/third-party moderation). It makes sense for Youtube and Google's reputation that they didn't get into it earlier on, but I still respect Youtube for the culture it has created, even before getting bought by Google. I don't know when it all started, but gameplay video subscriptions are a huge part of their content (I'm sure there are some good sources on this), and with the huge audience, content creators motivated by $$$ really early on.
The urge to share gameplay is something that has existed in everybody long ago. I myself was wishing I had better hardware for recording gameplay back in my early college years and uploaded a video 2006. I had also partaken in some live streaming of gameplay as far back as 2009 on different services.