Per the question, I used Minwax until it faded, but it was still quite obvious and visible (close to the last pic in the question post).
Then, taking a page from the "mayo and ashes" tip, I mixed 1-part mayonnaise (Hellman's) to 1-part baking soda (not baking powder; I used Arm & Hammer brand) as a cigarette ashes replacement. I rubbed the mixture along the grain directions using a paper towel and some pressure.
The stain faded, then vanished. The resulting wood is smooth, and considering the baking soda is abrasive, appears unscratched. After wiping off the mayo mix, my table may smell like a turkey sandwich, but it's beautiful again.
Now all I have to do is polish off all the Minwax buildup from my earlier cleaning attempts.
I would never have thought of going with the grain had I not spoken to the lady from Restore and Adore, and had the baking soda idea fell through, I would have gone with the "last resort" 4 0 steel wool with a lot of Minwax applied.
Pic of the repair:
The reason for cement board is to give a good adhesion surface and to increase stability. When a floor flexes from the weight of people walking, or from movement of the house, grout and even the tiles themselves can crack. For a temporary platform intended for a cat box, you're not looking at a lot of weight causing flexing, so I doubt there'd be much cracking. For adhesion, just check the product you're using to make sure it will adhere to wood, but I believe most thin sets and especially any mastic will work without issue.
Best Answer
Try using oxalic acid - you may be able to buy it at the local hardware store as "wood bleach"; it comes in crystal form to be mixed with water.
Pity, though, to take that marvelous color out of sapele! Good thing you didn't buy bloodwood (muninga) or redheart... 8)