When screws get stripped, you figure out pretty quickly that only the friction of the head against the substrate is what is keeping the screw in place. if you could magically remove the screw-head, the part would be removable, even with the body of the screw still in place.
The screw itself is expendable -- you can replace it with a different screw later.
The cylindrical barrel of the screw is under very little pressure and could easily be teased out later, or attacked with tiny pliers.
In car repair, they sell some things called "easy-out", that consist of a left hand twisted spiral tapered thing like a screw, but not sharp. You drill out part of the item to be removed, press the easy-out into the hole, and press while twisting to remove.

They also make a similar thing that has an easy-out on one end, and a small left-hand drill head (sharp this time) on the other end. Drill in reverse into the screw-head until either the easy-out starts to fit, or the screw head is obliterated and you are done. You may not need a power drill for these -- sometimes just pressing and twisting the drill bit by hand allows you to make (slow) progress.

This set of four must be very common, and in small sizes, because there were tons of picture to grab on google images. The barrels are standard quarter inch. I got mine from Sears in the US, but others outlets will have them as well.