Scarify and mulch these places. Aerate the entire area using a spading fork - break up the soil some but do not destroy the grass roots. West of the grass will come back on its own. Scarify, seed and mulch in the most heavily used area near the fire circle. Scarify, mulch, and anchor the mulch on a steep trail leading down to the drain.
Pits needs scarification, mulch, and anchor the mulch.
The kitchen has already been cleaned up. Mulching is needed. Aerate the compacted soils in the opening just to the south of the kitchen using a spading fork. Do this as they go into the woods. The objective is to hide access to the trail so people will not use it.