possibly if you are interested in doing this sort of thing, you will be interested in doing it with free software. in that case:

the debian debconf video team has been doing this for something like 15 years without the iphones and ipads, and without a limit of 4 camera angles, with 100% free software, and supporting live streaming (with simultaneous recoding to various bit rates) as well as just recording. they can fix unfortunate choices of camera stream after the fact for the final archived videos, too

they document their setup at https://debconf-video-team.pages.debian.net/docs/index.html. the live video mixing software (to choose which camera angle to stream at any given time) is voctomix https://github.com/voc/voctomix which belongs to the ccc video operation center https://c3voc.de/

i don't actually know what they use to push the rtmp video streams to the streaming server, but rtmp is very widely supported, since it's what twitch and youtube use for streaming video. obs studio can do this and is probably the most popular software for doing it; it's also 100% free software. https://www.digitalocean.com/community/tutorials/how-to-set-... says that you can also use ffmpeg directly

    ffmpeg -re -i "Introducing App Platform by DigitalOcean-iom_nhYQIYk.mp4" -c:v copy -c:a aac -ar 44100 -ac 1 -f flv rtmp://localhost/live/stream

https://trac.ffmpeg.org/wiki/Capture/Webcam#Encodingexample1 suggests getting ffmpeg input from your webcam with

    ffmpeg -f v4l2 -framerate 25 -video_size 640x480 -i /dev/video0

and i can successfully test my webcam with just

    ffmpeg -i /dev/video0 -pix_fmt yuv420p -f sdl 'hi'

but i think that integrating an audio stream is a little more hassle if you want to try doing it manually with ffmpeg command lines (details in https://trac.ffmpeg.org/wiki/StreamingGuide and https://www.baeldung.com/linux/ffmpeg-webcam-stream-video). if you want a gui just use obs studio