Interesting responses. Thanks, guys. I should have qualified my 15 out of 190 in the original post. I did something similar to The Bushmaster above. First, I taped part of an index card along the left and right sides of the press, extending about 1.5 inches above the base. After a few primers popped out the front side, I taped two more pieces of index card to the base, one just to each side of the primer arm.
Because I decapped and resized in one session, I noticed that I don't have to drop the ram very far to remove and replace brass. Next time, I might put a piece of tape over the slot in the ram, at least over the top couple of inches. That should force all primers to drop down.
Another thing I've noticed is that some primers come FLYING out of the primer pocket. Working the ram more slowly inhibits that, to a large degree.
The reason I asked about the two LEE cast presses is that I might consider getting one or the other. As Rod B noted above, a lot of residue is removed when decapping. It might be worth it to me to have most of that residue go down the tube instead of all over the press and me.