That's along the lines of what I was thinking. I think he used a 6" drywall screw which shouldn't have made such a big hole. With the number of gadgets for storage that are designed to hang off the tops of doors, or the god awful hinge mounted door stops, whatever is used as a fix has to hold the additional stress on the hinge other than the weight of the door itself.with such large screws installed i would completely avoid the toothpick and glue or golf t method.. ive tried them both and didnt work.
ive hung over 1000 doors, and repaired roughly 500. having such a large size screw into the framing has pretty much killed any chances of getting a new screw to grab any meat. i would either use a long drill bit and drill through the jamb right into the framing then using epoxy install dowling. or the other method which is more involved would be to use a "dutchman" wood patch.. this involves routing out a section of the jamb and installing a new peice of wood where you just removed a section of it. this is a more advanced technique and requires a router and special bits