How do I replace an old ladder's anti-scratch strip?
I would use a strip of UHMW plastic it won’t wear out and can be held in with counter sunk screws  I have 2 strips of this plastic on the bunks for my boat and over 25 years of use they still look great, no marking on the boat hull.

I would not use a self adhesive strip as with pressure they can move and that would not be good in this case.