Have a lot of so called magnetic bit holders. They hold the screwdriver bits okay.
But the screw itself falls off. Is there a screw holder rather than a bit holder available for the drill?

Just put a small dab of caulk or plumbers putty in the screw slot. They do make screwdrivers with a split end or collar that you release once the screw starts.

They do make holders as mentioned by ma064. I f you are an electrician, SlipStik(a Wire lube) works. Folded piece of paper will work on flathead.
Magnetic won't work on Stainless or Brass. Drop of saliva works on Phillips?