I just had 7 broken screws in a 1965 playfield. My solution was to drill on all four sides of the screw with a small bit, then with a punch knock the broken screw out. The hole left is large, obviously, but still smaller than the base of the post. I then filled the hole with a two part expoxy putty. It comes in small round pieces, like a tootsie roll, and you can slice off what you need. You knead it together, as part two is in the middle, until it's ready, then force it into the hole. Once hard it's sandable and paintable. I used the post a guide to drill a perfectly placed pilot hole in the centre, then it was ready to go.