There is a product called Leather Honey that we use for conditioning carriage harness leather, most of which is black. Great product, but it says on the label that it WILL darken light color leather. It colors evenly, but how dark will depend on the brown leather itself. New brown reins will look more used, going from chestnut to brown. Keeps my harness very soft, flexible for use.
You would have to try it on the saddle underside, then let it sit a couple days to see what your get. I rub it in with my hands, just needs Dawn dish soap to clean it off your skin. Hands are soft after, but does not remove your callous’ if you need them!
I am not a Neatsfoot oil fan, do NOT like how it feels on my skin. So I never use it on leather goods here.