Probably OBS

OBS is practically the only sensible choice. Some people still use xsplit, for whichever reason, but good for them

VMix popularity exploded during the pandemic. A lot of conferences became a blend of teams/zoom/Google and VMix.

Might be hardware based like a multi-m/e video mixer (blackmagic make cheap ones), or maybe more of a screen manager (like barco e2, analog way livecore). But, unless there are production requirements, vmix is much more likely. It's (now) proven, and much cheaper!

OBS can absolutely do it. There are other open source softwares that can do it.
I've seen people bastardise Resolume into something that looks decent.
There are some online studio systems so everything you do is virtualized. Streamyard used to be like this, till it was bought by hopin (I think it was hopin)

Not sure what they use (you could ask one of them.) Bue we've used OBS Ninja for similar things. As far as I remember it works well, has acceptable latency and everything is open source.

Edit: Obviously you have to put in some effort to configure OBS to your liking, make appealing slides etc.