I've been tasked to do research what it will take to get my ward up with the live streaming.
The closest I've found for what your wanting to find is the "PTZOptics Move SE" camera series.
Their literature lists it'll stream directly to Youtube and Facebook.
It'll also go to services using RTMP protocol. I don't know much about the RTMP protocol, but a quick search of Zoom and RTMP seems to indicate Zoom uses RTMP.
If we implement, I believe we'd just mount the camera in the rear corner of the chapel a few feet above the doorway, so it can be pretty much unseen.
I'm not sure if we'd be better off with the 12x or 20x optical zoom.
There is a web interface (so someone on the network with a phone or tablet can easily manage it), preset camera positions, and they'll even do auto-tracking, which I think could be handy for some of the stake meetings.
There is ethernet cable running within a few feet of the most likely mounting location so that part shouldn't be an issue.
I'm still working on the audio feed.
I'm thinking a 3.5mm extension cable between the amplifier and camera might be used or perhaps the 8 pin Mini-din RS232 input. I just need to measure distances to determine the best option.