When I mention “stain”, i personally use a trans-tin dye in black color moxed with distilled water. I brush it on the raw checkering and when it dries it looks kind of grayish colored. And then I oil it with sealer or finish. It darkens the checkering, but still allows the grain to show through. I used to use black leather dye, but it made it BLACK with no transparency. I got away from that.
When I mention using shoe polish, it would be for simulating old wear and dirt/grime. Usually reserved for if a small area needs to be repaired and I am trying to blend everything with the surrounding look. Like in your case.
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B. Dudley
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