Well, I don't see how you can use a screw extractor without drilling, flatlander. Notice that the set in your link includes properly sized drill bits to match the extractors.
Now, a moderately crafty machinist can take a right hand twist drill bit and regrind it into a left hand cutting bit. Sometimes, they will grab and pull out a stubborn screw. In any case, drilling the screw is quite likely to damage the thread. If it does, i guess you need either a new threaded plate, or a new pair of shoes.
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