I originally posted that the RCBS APS press mounted priming tool didn't work well for me.
After I posted that, it got me thinking that after trying it briefly a few years ago, I never tried to see what the problem with it was.
I went and dug it out, and stripped it completely down to component parts. I noticed that the priming rod had some buildup on it, like old oil. It was bought second-hand, so somebody dropped some oil in it. Gave everything a good clean, re-assembled, and primed 80 pieces without it glitching.
Looks like the buildup on the rod was making the spring sticky which prevented the primer punch retracting properly, causing the bind.