Have you looked at it from the underside side of the playfield? I would guess it's a fully threaded post that goes thru a T-nut and held underneath with a nut. Many people don't realize there's a nut underneath and wrench off the post. If so, remove nut, punch screw thru playfield and replace T-nut and post.

Is there a T-nut underneath or are you dealing with a threaded wood screw? With a small diameter screw the issue using a screw extractor is getting the left handed drill bit to bite in the center of the broken screw. Center punching a broken screw is difficult without grinding flat. If just threaded, you can punch out and depending on damage either then t-nut it or drill and install a wooden dowel.