Just had a look back through the photos and the binding appears to serve no purpose other than decoration as it looks like the top & sides are from the same piece of timber, therefore no need to hide any veneer joins.
As mentioned long ago, I would have sanded the crap off. Just need to decide whether you want to keep the more square edged look vs rounded over? Too many pointy extremities to get right if choosing to round over. Best way to go would be 120 grit on a cork sanding block sanding at 90 degrees to the top and back. The binding looks quite thin in places and shouldn't need to take off too much other surrounding timber. This would also help bring out any interesting end grain too.
Thinking about turning the random orbital sander loose mate. Back at work now....so will think about it.