While I like thing:3093890 and thing:526286 quite a bit, they are small and can't get a lot of torque on a screw, especially if it is stuck. I could have used a conventional screwdriver, but most of those either don't have a large enough diameter handle, or they are uncomfortable under high pressure, or both. I could use an electric screwdriver, but frequently with stuck screws, these don't give a good feel for what is going on and it is very easy to strip and grind out the head, making it impossible to remove the screw by normal means.

So I tried to make a bigger screwdriver head. Unfortunately, under pressure it either distorted the plastic or the bit just spun in the head. The solution was to add a cavity to the head and fill it with the strongest epoxy I could find (J-B Weld Steel Reinforced strength 5020PSI).

This has worked well in my testing so far.

1. Print the epoxy head and the plug. I tried printing the plug out of both PLA and TPU, both worked.
2. Lightly insert the plug into the hexagonal hole. The plug is tapered for ease of insertion and best fit. If you print in PLA, be careful to not jam it all the way in; it just needs to be in enough to prevent the epoxy from dripping into the center hole and in enough it won't fall out if bumped while mixing the epoxy.
3. Follow the directions on your epoxy; the cavity itself can be used to mix two part epoxy. It's probably ok to slightly under or overfill as long as the plug keeps the center hole clear. Mix well (try both stiring and spinning the stir rod). Try to keep the epoxy mostly in the cavity; it may help to tap the part on a hard surface to release bubbles. It might be necessary to wipe some of the excess epoxy off the plug with your stirring tool.
4. When done mixing, try to leave the part as level as possible so the epoxy settles evenly.
5. Allow the epoxy to partially set until it is no longer runny and pull out the plug before the epoxy sets so hard you can't get it out. (You might be able to leave a TPU plug in longer, as it is flexible and probably can be pulled out even if the epoxy is nearly hard.)
6. Once the epoxy has set, sand the epoxy surface if desired. Fit check your bits and trim the hole with a hobby knife if necessary. Note that the hole should be very tight, and short bits may be difficult to remove as they don't have much surface area to grab on to.
7. Allow the epoxy to fully cure before applying pressure. I let mine cure double the time in the epoxy instructions before trying it.

This openscad model is somewhat parametric, but it's sloppier than most of my models, there's a lot of hardcoded numbers in it and changing parameters is not well tested.

If your bit doesn't fit or is a different size, adjust hexw. The original thumbnut model also allows adjusting number of lobes, but this is not well tested in this model and other parameters may need adjustment as well.

The center hole has the smallest thickness walls I could get to slice correctly. (It assumes 0.4mm nozzle.) The idea is to have just enough plastic to retain the shape of the hole for the bit, but let the epoxy provide structural strength that won't bend or strip under pressure. I added a fillet at the base for strength so the center hole walls wouldn't break off while mixing the epoxy. There's also a thick lip at the surface with a nice rounded taper to make bit insertion easier.

The epoxy cavity also has a fillet on the bottom edge to reduce bubbles that might get trapped in a sharp corner. The cavity is oval just to make sure the hardened epoxy can't easily turn inside the handle. (Not sure if this was necessary.)

The "top" surface of the handle is rounded rather than flat so you can use the palm of your hand to press hard on the head (if necessary) to keep the bit from slipping out of the screw head. (This is especially helpful for Philips style screw heads.) There are some interior cavities under the rounded dome that are intended to force the slicer to add extra supports in this area. Because of the round, this was designed to print cavity facing down.

I sanded the epoxy surface on both of the pictured parts to make it smoother and improve appearance. It is questionable if it improved the appearance of the surface. I also slightly under filled my first attempt, and slightly over filled the second attempt. The under filled version worked; I'm not sure if the overfilled version looks better.