I find that I can hold the tap reasonably straight when I am just pulling a trigger and flipping a switch. Instead of drilling the desired holes, I ended up damaging the drill's clutch. After that abusive day, my clutch slipped with the slightest torque, rendering the drill just about useless. Normally, I would respond to the damage by taking the case apart and trying to determine what I broke. However, in this case the adjusting collar for the clutch fits over both halves of the case, and the collar can not be removed without first removing the chuck.
Reversible drills lock the chuck in place with a left-hand threaded screw inside the chuck.
The impacts seem to loosen stuck threads very well. The problem with my drill was that my clutch was damaged and I couldn't get enough of an impact to loosen the threads. I had to get to the clutch before I could get to the chuck. Fortunately, the plastic case is flexible enough to allow it to be pried apart enough to get access to some of the internals, even when it is held together in the front. The first thing I tried was to jam a screwdriver into the reduction gears and attempt to whack the Allen wrench to break the stuck threads free.