If the surface you're tapping is flat, I would just make a tap guide that sits on the surface, like every machine tool student makes in their first semester. If that surface isn't flat, but the opposite surface is flat, then I would put the tap in the drill press and tap by hand with the spindle unlocked/free. If you have a milling machine, and are laying out your holes to specific dimensional locations, I would position them on the mill and use a center in the spindle and an appropriate tap handle to keep them straight.
I wouldn't do them under power outside of a CNC myself. Too much hassle the time it doesn't work, and even with a hole popper at that size you're likely scrapping the part at the worst, or modifying it to the next larger thread size at best.
H3 is the class limit for how far oversize from nominal pitch diameter of the thread (which maybe is what you meant by depth). H3 is .001-.0015 over nominal pitch diameter and fairly typical in those sizes. H3 threads assemble easily without galling or binding.
If your setup and tapping method leads to threads that aren't sufficiently tight in fit, one way to deal with that is to use an L class tap, which is undersize relative to the pitch diameter.
