I have not seen the project, however, I am assuming the island has at least one receptacle that satisfies 210.52(c)(2).
My question is about (the visible side, the finished wood side) of the island. Does that count as a wall space? does 210.52(a)(1) & (a)(2) apply? I think it does.
The "wall" of the side of the island does not count as a room wall under 210.52(A)(1) and (A)(2), and even if it did its whole length would, IMHO be
interrupted by a fixed cabinet, and so not count.
So that leaves only (C)(2) which others have discussed.
Now if the island had been instead just a partial divider wall between kitchen and the adjacent room, there would be more agreement that it would be a wall under (A). And it would then need outlets on both sides.
However, that is just logic, and the wording of (A)(2)(3) seems to imply that the face of the island may have to be counted if it is a "free standing bar type countertop" but not if it is instead a
kitchen countertop with its separate rules. The fact that the far side of the island is in the kitchen and is not set up for bar use in a non-kitchen room seems to be relevant. It may come down to whether the structure is an island, surrounded by kitchen or a divider between two rooms. From the photo it seems more like the latter, but we need to see more.