IMO, driving them in deeper in the face of the door will be difficult at best. To do this, which is possible, the door needs to be solidly backed so it has no "give" when the screws are driven in. That said, if you are successful in doing so, the door material will still need to be carved out so the soft door surface that is swelled out around the broken screws can be cut back to solid material so it can be filed with a filler or putty of a sort.
My suggestion would be to use a Dremel tool with grinder, right angle grinder, or attachment for a drill to remove the metal that is above the door's face. Doing this, you will also remove the soft material that is above the surface so it is ready to take the filler needed to restore the finish of the door.