Looking at on-line pictures the “collets” look more like tap carriers than true “collapse to grip” collets.
I have a small clutchless tapping attachment of unknown breed that uses this system. The holders have a socket grub screw ti grip and drive the taps. Taps have to have a flat ground on them for the grab screw to grip. The square isn’t used. Drive is via a dog clutch arrangement with a spring pulling the head proper back to engage the clutch. Follow the self feed with the mill or drill quill, when you stop feeding the tap goes about 1/8″ deeper before teh dog clutch releases. Came run a box with some other stuff including some modified taps. Gets occasional use when what I have suits a job too small for my Pollard heads.
If you have nothing better the system could probably be used on that Tadco head. Several pictures show a lot of collets in the spacers provided in the box. Many more than collapse to grip collects would need over that range.
Clive