If something like this happens to someone else, for future reference, you can simply purchase a screw extractor. It's essentially a screwdriver with teeth on the end (instead of a flathead/Phillips/Robertson/etc tip) which grips the broken screw shaft. You just unscrew the broken screw like you would an unbroken one. You might have to finesse it for the teeth to get a good grip on the broken shaft, but it works well in most situations.

Did you (or anyone else) ever find the answer to that question?

I don't need to replace a broken screw, but I'm wondering about an adaptation that might benefit from using a screw of a different length, so diameter & pitch of those screws are of special interest.