Ok, first I have to ask how hard are you torquing these things down to snap them? I see stress cracks in the clear around the ball guide screw too. Ease these in...
That screw looks way too small to try to center drill and use a a standard extractor on.
Now. What I would do. I’d buy a screw extractor from Rockler. It’s a pain in the arse to use but it drills *around* the broken screw. I use these when I can drill a new hole all the way through. I then use a post I can secure with a locknut underneath. Far more secure against angry pinballs.
*extractor disclaimer. Much easier to use if you make a guide block to drill through and clamp it down to the playfield-examples of this are online- and these things are brittle. If you screw up and hit the metal screw, it will break some teeth. If you don’t have a Rockler nearby, order two.
And note they have several sizes. Pick the one that works for you.
F2AC97AA-4C0C-4812-AEB8-21561E351A2E (resized).jpeg

I suggested it because I restore dozens of playfields a year and use these whenever I run across a broken stud topside, particularly if it was broken by ball contact. Using a locknut/washer under the playfield makes for a more secure post when the job is done. What type of ugly patch are you talking about? This method drills a hole through the playfield that you then fill with a new post and secure from below. I’m not talking about using it for under playfield screws, there’s a hundred other ways to skin that cat.
That said, I followed your link and am willing to try that method next time if I remember it. You can never have too many tools.