I used to buy fitted sheets until I realised that their fit was dependant upon the thickness of the mattress, and they would always be a poor fit on some beds.
Plain sheets stay put by being large and the excess material being weighed down by the mattress, so they just work.
On replacing elastic, I having a sewing machine, but am not sure I would bother to replace fitted sheet elastic. I would definitely not attempt the job by hand because hand stitching is weaker than machine stitching.
On what to do with surplus textiles, I cut them up for cleaning rags. A stack of cotton rags is a useful thing to have in the bathroom where surfaces need to be cleaned regularly. These rags are useful because they are tougher than a cleaning sponge (or the commercially available cleaning cloths) but not as abrasive as a scouring pad. I wash and reuse them until they're worn out, at which point they can be composted. I keep another stack of rags for cleaning bikes.There it's useful to have some thick fabric rags and some thinner ones.