I removed the tub drain overflow plate to reconnect the trip-lever lift assembly (which had actually fallen off the lever) but in the process one of the two screws holding the plate snapped off in the overflow pipe due to rust corosion – how do I get the broken piece of the screw out of the pipe so I can replace the broken screw?
Normally we replace the entire tub waste if it is that corroded.
You can try drilling out the old screw very carefully and retapping the new opening with the next size larger tap. Or you get what they call a large cock hole cover (a crome plated cover) and attact it to the existing over flow tube and buy a rubber stopper for the outlet.
The cockhole cover for the tub can either be a flat plate type OR the better one is the one with the over flow provisions.