So far I’ve tried Screenflick and Camtasia, which both work fine, but both can only do 30 or 60.

Doesn't have to be an editor as well. I have FCPX or Premiere for that.

 

I'm pretty sure you can only record integer divisions of the screen refresh rate: 60, 30, 15...

You'd need to do a very long second pass to re-encode to 24. Not 100% sure but almost.

 

I don’t know a whole lot about it, but OBS Studio for Mac is supposedly really good and used by many of the music producer Youtubers I watch.

They are sponsored by Twitch and Logitech and the software is free.

 

This is correct and it is logical as dedicating resources to recode the frame rate could cause an unresponsive system while recording.

If you are going to for quality and want to capture a lot detailed motion, I would record at 60FPS and use your NLE to achieve your target frame rate.

 

We shoot all our video at work at 24fps. Now they want me to insert some screen capture b-roll into one of the videos, and I know combining 30fps and 24fps on the same timeline is bad news. Possible, but I don't want any skippy frames or hiccupy look.

I'll try the 60fps thing and crushing it down to 24. Or, I have Aimersoft Video Converter. Maybe I'll run it through that to see how it looks converted to 24.