I wonder if someone makes a reverse threaded chuck that would self-tighten when spun counterclockwise. That would make an impact driver a good prospect.
An oscillating multi-tool would be easier to flush-cut with than a typical demolition saw although some demo saws have a way to mount the blade for flush cutting.
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A socket wrench type socket is available in multiple sizes, has an inversely tapered socket with sharp, tapered inwardly bolt/screw catching Easy Out type holders for extracting bolts, screws. I have two sets. Absolutely indispensable.
Or Vise Gripes?
Or 4" pipe wrench.
If you buy a multi-tool, get one that has the "tool-less" blade change feature.
In considering whether to remove or cut-off a screw, you obviously should consider whether a new screw is likely to hit the remains of the old one.
Tommy uses something like a core drill at 1:30 in this video:
https://www.youtube.com/watch?v=Gz1SatVZ-zE Getting the core out of the bit might be tedious. Perhaps lubrication makes it easier - or some special technique ?