Holy bologna. (Side note: Now I kinda want to eat some bologna again.)
I had been wracking my head over how to get animations in iOS to not completely interfere with the interface when I don't want it to.
That's it. I had been doing the animateWithDuration:animations: block and it had been stopping all user interaction. Doing some research I found the old way using beginAnimation and commitAnimation did not cause these pauses. Which is great but as that is the old way, I needed to get something better than that.
Turns out there is animateWithDuration:delay:options:animation:completion that you can shove UIViewAnimationOptionAllowUserInteraction into as an option and voila! The animation doesn't interfere at all. I have no idea how I missed this!