Snapped screw remover, an opportunity?

[Jacob]

....- up to painted wood getting a lick from my small pencil torch (same size as a creme brulee caramelizer, coz that's what it is), .......

That's a good idea. I often use a blow lamp but I've got a little culinary one ( for singeing orange peel for cocktails etc.)
The main thing is to clean the slot/cross, use sharp fresh screwdriver and the magic trick is to tighten a click before you start to loosen - it breaks the seal and if it damages by camming out it doesn't matter as the other side of the slot for loosening will be undamaged so you've got a second chance and you know how tight it is.
Another thing is to have exactly the right screwdriver or bit. A slot needs a bit slightly less wide than the slot, so it doesn't ride up the countersunk sides of the hole.
A long screwdriver helps - it's easier to keep it straight and perpendicular to the screw head.
Brace and bit not good - if it needs that much force it'll break anyway.

 

[HJC1972]

I find that sometimes merely giving the screwdriver a tap with a mallet before loosening has been enough to break the rust bond on old screws. Failing that, i always do as the previous poster has mentioned and tighten them up before trying to loosen them. Also, I have a big old beast of a soldering iron and this has got me out of trouble many times, particularly on wind-out Victorian table tops, when not only has the screw rusted in but the top may have contracted/expanded, thus twisting the shaft of the screw. It’s a right old PITA waiting for the screw to heat up enough though.

I have also managed to get them out by plug cutting round the top and then you can sometimes just get enough grip on the stem with a pair of narrow nosed Mole grips.

I do like the idea of a long bored and narrow hole cutter though. That could save a lot of head scratching.

 

[HJC1972]

And another thing....

I have bought various types of damaged screw head screw removers over the years. Usually along the lines of a reverse cut drill tap or drill bit. Aldi had them in today when I called in.

Useless, every single one, every single time I’ve tried them.