Migrated to:
https://www.printables.com/model/547974-drill-stand-soldering-iron-holder-for-heat-set-ins
I created an inexpensive heat-set insert press with a commonly available soldering iron, heat-set insert tips, and a cheap drill stand (to make a cheapo drill press substitute).
I improved upon Thingiverse user papa50's design, which only squeezed the iron in the drill clamp, by adding a rear closure which prevents the iron from backing out of the clamp while under pressure.
I recreated papa50's design in Fusion 360, then added M3 x 8mm heat-set inserts to the base part, so I can screw down the rear closure (I used M3x16mm screws with integrated washers and lock washers from a hardware kit).