I agree that I wouldn't tap anything into the screw head but using an easy-out should not involve having to tap it in if the right size easy-out is used.
The taper of the tip of the easy-out is likely extreme enough that the flutes will bite into the sides of the recess when the handle is turned counter-clockwise. I'm not guaranteeing it will work but it's an inexpensive experiment and won't make the situation worse if it doesn't.
The taper of the tip of the easy-out is likely extreme enough that the flutes will bite into the sides of the recess when the handle is turned counter-clockwise. I'm not guaranteeing it will work but it's an inexpensive experiment and won't make the situation worse if it doesn't.