the teckniks heads you show at your link are really a floating tap holder in the profile of an ER collet (ER-25? ER-32?) the end of these on the I.D. are rectangular with one or two grub screws to lock on the square at the holding end of the tap. (the square of most taps does not vary per size. It's the same rectangle for one range of sizes, then steps up as tap size increases. Where you could have a hundred different tap sizes, the rectangular size is only about 3 to 5 possibliities for the range of taps these machines accept.
Not very quick change at all. But they can use standard ER collet holders, which is a benefit.
The jacobs style rubber collets you have are excellent in that they are very very small and use a much smaller "head" than a similar range ER Collet. Quick change they aren't!
A bilz type quick change head would take up a lot more clearance in the work area and would create interference problems where the jacobs type would not.
A simple ER collet extension with an ER head, that would accept the techniks ER collets would be a good thing for main spindle tapping. Just mount it like a boring bar on the turret.
Solid tapping has been commonplace on turret type Mazak turn-mills for perhaps the last 7 or 9 years? Prior to this technology becoming standard, driven tools had to have float, just like older m/c's before solid tapping.
SOLID TAPPING at the lathe spindle is still an option to this very day! It is a seldom bought option, however. So you still need an old fashioned floating head if tapping on center using the lathe spindle. However, if you can find a cylindrical shank collet extension that accepts a bilz style head, you will be in biznez for floating taps at the main spindle, using a fixed/lathe style holding arrangement at the turret.
Lastly, if your taps are all the same brand, sometimes there is an adjustable backstop inside for the tool to locate on axially (to the tap) when it's pulled down by the action of the ER collet. This has some decent repeatability, so once you dial in the offset of one tap, it can be swapped out for the exact same brand/part number WITHOUT the need to re-set the tool. The length of the tap is, in concept, a "pre-set" tool!
-90% Jimmy, the Mazak internet fairie