That's interesting. I've heard of that style of collet but never really had an appreciation for their implementation. So the square broach prevents rotation of tap to collet. But if you don't have tang receptacle then the possible slip will be on the O.D. MT tapered arbor surface the mating quill ID. I'm not sure, maybe taps come (or came in) nominal shank sizes, so maybe 3/8 shank covered several taps on either side, then set down to 1/4 for smaller or step up to 1/2 for larger range?. That would make it more universal & maybe less collets to own. I know with the tap holders I made the shanks tend to vary a bit by manufacturer & metric are different again.
I find the whole tapping on a lathe exercise kind of variable, people do it different ways. I'm a bit leary of driving in under power because you have no feel for overload unless the tap spins in whatever is holding it, or worse yet breaks. For small stuff I just lightly push the tailstock into the part, either rotating the chuck real low rpm or by hand. If its a higher load I just initialize threading so engagement is straight & attach a tap handle so I can back it out & feel what's going on. But I am certainly no pro so this might be making more work than necessary. I often wondered why there wasn't a clutch type tap holder similar to what you see in a (mill/drillpress) tapping head, except for tailstocks which is now the stationary part. Maybe they exist?