This kind of problem is very common. The crack was glued without clamping or finishing. That’s why dust or sweat was rubbed into the cracks every time someone touched it. Even the glue was dirty.
The crack was across the grain, so it wouldn’t be an invisible repair even if the original parts were reused. So after I removed the damaged part by about one millimeter deep, I installed a thin maple veneer. Special care was necessary so as not to damage the original “Contour Body” decal. The old Fender guitar decal is usually placed right on top of the finish. So if you use masking tape to protect it, you can cause serious damage to it when you remove the tape, because the thin decal can be pulled off.