I've had that flaking insulation problem too. It can be annoying on the pickup wire, but I have a reasonable workaround. The problem is not so much the absence of insulation as it is the possibility of the wire moving and coming in contact with the frame to cause a short.What I do is make a tube around the wire with insulating tape and shove as far as I can down the wire toward the pickup shoe. I'll shove a wad of insulating material anywhere that seems necessary to keep that wire out of contact with ground. Messy, but it works.
Another easy fix that I've used is to cut the wire where it becomes accessible as near as possible to the pickup shoe. Clean the flaked insulation off as far as you can reach with a small knife blade. I then take a length of insulating sleeve and push it over the wire towards the shoe as far as possible. You'll be able to push it past any possible contact with the frame. Then you can solder a piece of replacement wire to the stub of the original.
