Vintage 4 sided paddle strop

I bought a 4 sided paddle strop in pretty good condition off eBay. Lady even felt she charged to much and refunded 2.00 to my paypal account, and considering I only paid 9.99 for the strop thought it a pretty nice deal.

I took it apart and have been treating the leather with Neats-foot oil. It was rather stiff in the beginning, and the oil soaked right in. Kept it up for about 3-4 days and last time, I think it's good to go now, since the oil stayed on the leather a little longer before soaking in. It is not softer and more pliable and the shape of the leather in one piece, about 1 1/4" wide rectangular, and all one piece. The wood is in good shape and the handle screws in to the metal which tightens the whole of the leather. One side of the wood has red pigment and I sanded that fairly smooth. The opposite side had some kind of coating but when I started sanding a lot of it came off and the surface looks kind of brown and fuzzy in places.

Would I be just as well to go to Hobby Lobby or other and buy a thin piece of balsa wood and glue it onto the side that is fuzzy, taking off the last of the peach fuzz? The Red Pigment side and opposite fuzzy side are exposed even when the leather piece is on. Leather tightens up real nice. No cuts, notches or bad spots on the leather.