Try smaller pastures and more frequent moves. In high-intensity cattle grazing, which mimics natural herd behavior as closely as possible, the cattle are placed in a very small area, so they're forced to eat the entire area evenly, but they are moved to fresh pasture every day. Each pasture needs to rest for 30-60 days, depending on speed of growth. If the area is large enough that they can be picky, they will, as you've seen, trample and overeat some areas while letting others go uneaten.
It isn't strictly necessary to have "proper" feed for them. Ducks do fine on chicken feed, and in my experience, geese do too. I feed my geese some layer feed and some milo/sorghum. I think grower is better than layer feed for geese, since they don't need as much protein.
Do you know what types of grass/weeds are in the pasture and if they're nutritious enough to sustain geese? We have a variety of weeds in our yard and the geese primarily stick to just a couple different types.