drillrepairscrews
I saw this once years ago, but can't find it now. It's a drill bit that drills a conical hole whose taper matches the cone left by a pencil sharpener.
The idea is that you can repair stripped screw holes by reaming them out with this drill bit, and then plugging them with a dowel sharpened with a pencil sharpener.
This seems like a really elegant way to repair a stripped screw hole.
Ideally, hole saws have holes or slots in their sides, so you can pry out the scrap, or in the top so you can push it out.
(source: mkmorse.com)
Poke in there with a screw driver or other sharp object. You shouldn't need to disassemble everything; just unplug the drill.
I assume yours don't have this already. If you are going to be doing this a lot, you might want to buy a new hole saw with better holes.
Alternatively, you can try drilling holes in the top of yours. Secure the hole saw in a vice or otherwise clamp it down firmly, and drill two holes on opposite sides in the top.
With two holes in the top, you can just push out the scrap with two large pins or nails.
Edit: some have larger side holes which may be easier to clear out. You could possibly enlarge the holes on yours with a file.
An impact wrench will do it if the head is not stripped like the one in the photo. As noted , keep high down force to prevent stripping the heads. My Makita has given excellent service.
Best Answer
You may have seen a taper reamer, there are a few different tapers. They are made to cut steel so wood is easy.
They would need a 1/16 to 1/8 pilot hole as they do not cut in the center. I have a few from when I worked in a bit factory. I expect they are very expensive to buy.
However, I would drill a regular hole and use a dowel.