I use a spring loaded center (shop made) also. I've been using a piloted tap wrench lately (picture below). Bought a pair that goes from 1/16" - 1/2" off eBay from a seller in GA for about $40 delivered. I use the piloted tap wrenches in the drill press and vise also.
I just use a standard drill chuck -- just for the reason you pointed out, the the tap will slip when it meets the bottom of the hole, or when the resistance gets to much (taps are hard and wont mar like a drill bit). I do have variable speed and this is done very slow. At the first slip, I reverse and will usually try again and get a thread or two more, if needed then I finish the threads with conventional means by hand. (use lots of thread cutting oil!)
I have to admit I power tap on the lathe sometimes. Just did 4 x M10-1.5 pieces in 6061 over the weekend... with a hand tap. Usually I use machine taps, but I didn't have any that size. Made me cringe, at first, but I did like you except when I reverse it out I add a drop or two of TapMagic and it usually finishes the hole. The stuff really is amazing. I use the all metals formula. Never have tried the one that's "formulated for aluminum."
I have been thinking about getting another Jacobs/MT taper for the tailstock on my big lathe and mounting a tapping head. Then I can count on the clutch in the tapping head to stop me from breaking taps. I usually tap at about 70 RPM (the slowest gear speed on the 1440). I don't power tap on the baby 7x10/16 as it has nearly no torque at low speed. Even in low gear. The 8.5 x 18 is belt drive so I guess I could reverse the pulleys and get good torque at low speed, but its a dedicated collet machine for small round stock now. I do tap some teflon parts I make on it, but teflon takes nearly no force, and its self lubricating.
I've never had a problem power tapping in the lathe using a drill chuck. Tighten the tap a little tighter that normal for a drill of the same size. No cheater pipe used. Use a good cutting oil like Mobilmet 766 and get after it. If the tap starts to spin, I stop, back the tap out, clean chips out, go back using a tap wrench to finish up the job.
I have one of these. It works great. It also has a gizmo to hold dies. The holder slides on a rod so you can feed it forward. If it gets stuck, it just spins in your grip. I don't know that I've ever had to use the bar that goes in the hole. The knurled barrel provides lots of grip. Mine just chucks in the drill chuck.
Brino, they are not shop made. I purchased them. They shouldn't be that hard to make. The hard part would be attaching the tap chuck to the outside tube.
My tap holder fits in a drill chuck. It is limited to 1/4" and smaller taps.
I couldn't find it online and don't remember where I got it.
(Memory is the second thing to go. I don't remember what the first thing was.)
The Die holder uses a M2 taper. I got this at Little Machine Shop.
These are pretty useful and available at many sites. Large and small holders with a square hole for socket wrenches.
You can also stick a hex socket adapter in them and chuck them in a cordless drill with a hex to square socket adapter.
My most used method of tapping is by chucking a tap in my cordless drill and setting the clutch low. I've never broken a tap with that method. It is good for quickly backing the tap out to clear it. I never had much problem starting them crooked. I'm pretty good at holding a drill at 90º. Some people aren't so that might not be recommended for them.