Multiple screens is not a string requirement, but it will make things MUCH easier. You can manage the stream and (try to) keep anything private on the screen that is not shared as a source in OBS.
I use OBS and hooked it up with restream.io. I don't recall the exact process but it was fairly straightforward after signing up for restream.io. Youtube seems to keep them recorded automatically, I don't think I had to do anything extra.
Restream.io can in turn connect your twitch, youtube and other platform accounts. All of them will go live together (if everything works)
OBS "Scenes" allow you to build up other things on your view like the chat box, other camera feeds, and any static elements you want. You can setup multiple scenes and switch between them if you want.
I used the discord stream kit chat overlay from here: https://streamkit.discord.com/overlay there are a few configurable options. But not overall size. I ended up writing a little javascript bookmarklet to manipulate the size: