I have worked with a couple of 'forest schools'.
It is noticeable to those like me, who are occasional visitors to the sites the groups use, that over time the flora is damaged, the land is marked by footfall traffic, paths are very evident, debris is 'cleared' creating more of a 'well used' space. The use slowly erodes the understorey, slowly compacts the earth around favoured routes and spaces. Compaction damages the soil structure, the micro life that inhabits it, the roots, the mychorrhiza, etc.,
It all slowly adds up, even if the group is infant school children. You can see it in extremes on public rights of way, on forest walks, hill paths and where folks gather 'kindling' for the wood stoves that are in fashion.
No idea about this specific site, or the group, but that's my tuppence ha'penny worth from noticeable differences in sites over time.
Maybe, since the site is a SSSi, that it really is time to take stock and think hard about constant (and by the sounds of it this is at least very regular) use of scheduled land.
M