I will say, if you have access to the correct equipment, spot or centre drill, carbide drill, and easy-out, or retap it once drilled.
If you are really lucky, often when you drill it, you get a nice little coil you can pull out with pliers / grips. You MUST set it up and drill it dead nuts centre.
Iceng is correct. The only other item I would look at once they are out is the threads in the part. Something is going on there, either with the screws or the thread dims in the hole. Regardless I would take a serious look at chasing th threads with a quality tap. By quality I mean something other than hardware store Irwin brand crap. A spiral flute bottoming tap would be good.
I was wondering about the loctite too. If it was loctite I would still chase it to remove the residue, but that's just me. I want those threads as clean as possible before I ran another screw down a hole that I had to drill a screw out of.
100% be chasing that thread regardless, because it a broken bolt occured in it, it may habe been a 6mm bolt in a 1/4 unf hole. Fun times. Those suckers go in like 3 turns easily before they even start to resist.
maybe not on the beaten path, but have you called a few machine shops or mechanics near you? Most mechanics I know are pretty well versed in extracting a broken screw..but we're in the rust belt.
Its on its way to d wilson. The extractor is really jammed in there and don’t want to trash my ATX or make this worse than it currently is. Got it back from a cerakote job and feel like the paint may have gummed up the threads.