I went with random mixing to add a little variety to the experience. I know personally when I'm at a party I get stuck in a rut and end up talking to the same people over and over. A tool like this where you're warned when you'll mix but you don't know who you might end up talking with next helps with meeting new people.

I also figured it'd be easier to add features like self-sorting later on after I proved out the random mixing idea.

I mentioned it in the other comment but yes, I'm using Twilio's Programmable Video API https://www.twilio.com/docs/video