A clever and expedient technique but turning the chuck on my mill by hand doesn't work well with my 80 year old hands.
Like you, I decided that the best time to tap a hole is while it is still aligned with the axis of the drill that made the hole. The pointed tap wrench guide thing was invented by a mutant with three hands. Piloted tap holders are commercially available but they eat up too much headroom when installed in the drill chuck. My solution was to make low profile tap wrenches with a built-in pilot that could be grasped in the drill chuck. A number of variants of this idea are described here...
Tap holders
The simplest of these has a fixed size hole to hold the tap which is secured with a set screw bearing on one flat of the square drive. Smaller taps have the same shank size...
Standard Tap Dimensions (ANSI Shanks) - Tapmatic Corporation
so this fixed hole size works well. For the larger taps, shank size varies so I made a tap handle that accepts collets to accommodate the various sizes. Since I knew I would need to made a number of collets, I chose the simplest design I could imagine. A bit of steel rod is drilled to a sliding fit on the tap shank. Then a bit of the rod is cut away to allow the setscrew to bear on the square drive.
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