We did a theatre a few years back with camstreamer. It's an app that runs on an axis camera - no other gear needed. We had it set up to start a private youtube stream. It does facebook and other platforms too but we never tried those. It was solid except the camstreamer app needed updating a few times to keep up with the changing Youtube API. I left the company so I don't know that continued to be high maintenence. Maybe it updates by itself now or another solution does, something to consider.

There was a way to make it super simple where you could flip a switch linked to a contact input on the camera to start and stop the stream. You could have a "on air" light too. Never tried that though, we just had the user login to the camera to control the stream.