I recently bought a used JCL MC that came with a black OEM alligator strap. The strap is about 6 years old and is in good structural shape but the top surface of the light gloss finish had worn in areas, leaving much of the 6pm side of the band looking dull, probably owing to day-in and day-out rubbing on a shirt cuff. So I spent more time than I should trying to figure out how to address this, even using some high-end Saphir products that I use on my dress shoes, only to make it worse. Now instead of a dull black, it was drying out and looking grey/white in splotches. Polish wouldn't stick (cream or paste) and I was getting ready to chalk it up to a total loss. Then I tried something I thought might result in disaster but actually worked well.
I took a container of Kiwi Heel & Edge, black, and squeezed it onto a QTip and then painted the top surface of both ends of the strap (noon and 6pm) and voila, my strap has been restored to beauty. I carefully painted and it soaked in, leaving an even surface allowing the grain to show through. I have done this before to repair a scuff on the right heel on a pair of black dress shoes and it lasted for a few months and then I needed to re-apply. I know this is probably not ideal but I wanted to share it as it really did work and I couldn't find any other solution on this forum or anywhere else on the interweb.