We can do things like run nginx with nginx-rtmp-module as a proxy, but then you have to put stunnel or something in front of it for Facebook because nginx-rtmp-module doesn't support rtmps (and development seems completely stalled).
So lots of reencoding happening on the fly, which is why I know of some streamers using dedicated streaming computers, that take the OBS stream as input, but I don't know what software they use to distribute it.
Being able to do this is actually the number one reason why I strongly believe nobody should ever become a Twitch affiliate.
Actually, ffmpeg is the other solution I was looking at playing with on the weekend.
I just haven't yet found a good way to export active stream status from the CLI out to other interfaces for the tool I'm working on.