You photos show a Finished Leather, not a pure (dyed) Aniline. Finished leather is painted, and your paint is wearing off from use and friction. Yes, the correct way to do that is paint the edges. You can go to craft store and buy your own paint and shade to color to apply (less than $ 10), or have a pro come in and airbrush those areas ( $ 200 and up). One repaired, keep that area conditioned with a leather cleaner / conditioner to slow down the paint loss.