That was my first thought too but unfortunately most of it isn't on the surface it's been drawn into cracks inside the binding. There are still a few small spots where glue on the binding has absorbed some but that is a pretty easy fix like you said.
My Keda wood dye finally arrived so after a little time spent tinkering with colours I found something I was happy with and stained the top, which means one step closer to finishing. I still need to put a burst on the headstock to cover up some spots where stain from the sides bled onto the face, and then on to clear coats.