Let's establish a few things.
Your hinge broke, because your hinge is made of cheap zinc that's chrome plated.
Here is how you remove it: securely grip the broken hinge with vice grips. Use a quality impact driver to remove the screw. That's it!
The advice on ez-outs was given by, well, idiots. This screw is not frozen in a steel engine block, not rusted by exposure to sea water, not secured in brittle cast aluminum. No, it's in fiberglass with some sort of soft metal stiffener. LH drills are also overkill.
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