I’m sharing this shortcut in case it’s helpful for other teachers. It randomly presents names of students to call on, working through the entire list without repeating.
It took me a couple of days to figure this one out. The key turned out to be shuffling the list itself, rather than randomly selecting a student, then removing the selected student from the list of those available to be called on.
Thanks for sharing! I did something like this once, too. Mine was Keyboard Maestro-based and didn’t have your no-repeats fanciness though.
This might be a neat tip, though: I was using a Logitech Spotlight remote. You can customize what the buttons do. So, I set one of the buttons to keystroke some ridiculous combination and assigned that to trigger the KM macro. This let me lecture away from the desk, but I could trigger the random-student macro at any time, so students had to stay on their toes.
Great tip, thanks!
I tend to wander the room with an iPad Air projected to an old (3rd gen) Apple TV. I use Notability as a whiteboard. As long as I’m in presentation mode, students will only see Notability. 
That is a cool set up. Are these institution-owned Apple TVs or do you bring one in yourself?
It’s institution-owned.
My college doesn’t actually provide them, but several years ago our then CIO had gotten a bunch for a department that was experimenting. I was in a meeting with him one day when I mentioned how much I hated being tethered to the podium. Since he had an extra, he gave it to me.
Many of our classrooms now have a Crestron AirMedia setup. Not all of them are in working order, though, and even for those that are — well, let’s just say that the only words I have for the AirMedia iOS/iPadOS app are words I was raised not to use.
I’m much happier with the ATV.