A couple of weeds growing in the soil does not equate to dense turf growing conditions.Nor is it impossible to grow dense turf in shady areas IF you select the right turf.(bluegrass and rye are not the right grasses). He states that he's got "lots of shade" and water penetration issues..I'm guessing he doesn't know the details of what else may have lead to the turf decline (like soil fertility, fungus, improper irrigation during sod establishment, etc.) but he needs to start with improving the soils ability to retain, not runoff, or fertilization will just run off as well.
Since he didn't state what turf type he has either, but that it was sod, we can assume it's a blue/rye/fescue general grow based on his area. There aren't many sod farms in full shade, so you can bet it's not a shade type grass selection.