After switching to smaller M2 screws, I put a board in, pressed down and flashed it. It worked on the very first try. No more soldering wires to pads.
Using it showed a few things to fix. The printer couldn't make the thin pegs meant to line the board up, so they became holes with a spare pogo pin pushed in as a perfectly straight metal peg, and one turned out to be enough. A raised FMW or TEST label tells the flashing jig from the testing one. And since a tug on the cable pulled straight on the solder joints, a small clamp now pinches the wires inside the base before the cap goes on.
The jig is done. Next up is the part I've been waiting for: the button's actual case.