Before, I could easily join public hangouts. But now, whenever I click on the play button it's opening a Youtube live stream.
A hangout is indeed a public live stream video that is shown on Youtube, are you mistaking it for something else?
Edit: What you are seeing on that page is called Hangouts on Air. You can't join such hangouts, only watch them. You can, however, join public hangouts that don't do any broadcasting.