All I can think of is that the reservation window isn't accurate or the sites are available for drive-ins, not reservations. When you look at the site list, do the little boxes have a 'R' (reserved) or an 'A' (available) in them and what color are some. The non-reservable sites are brown.
Probably best to call Reserve America and have them look at the site after you were assured by calling the campground that sites are indeed empty.
It's hard to believe that the campground would be empty all the time because folks can't get through with a reservation. Someone is getting into that campground - either reserving or just driving in.
What campground is it?