John Phillips didn't say how deep his sheared-off screw is, or what the
accessibility happens to be.
The piano action is easily removed, of course. Then there are about
three kinds of damper flanges I'm familiar with on uprights. Two screw
into wood, which is what I think he has. He says the flange screw broke
off and doesn't want to heat it up to remove it. I would have to ask,
why? Heat is about the best way of all to do it.
Stand a very short snip of welding rod on the broken end of the screw
after you have ground it flat and wetted its standing end with solder
for good conduction of heat. Use a micro-torch on the other end of the
welding rod, supporting it with a little jig whose tip keeps the rod
spring-loaded down against the broken screw.
When you see the wood start to burn around the top of the broken screw,
then you are doing some good. If not for the burn, you couldn't get your
long nose or a screw extractor in there, anyway. I like long-nose vise
grip pliers when I can use them.
If you don't want to extract the screw, do this:
Fashion a flange plate from some scrap metal like brass sheet. Fasten
it to the old flange by cutting a slit through the old flange to support,
then tapping the sheet and using a small machine screw to fix it tight
(glue last, also). The brass overlaps the neighbors and you tighten it
down with the two flange screws on either side.