Absolutely, it's the lack of respect thing that is the damaging aspect.

Incidentally, if NT are saying they "did give permission for access" that's being a bit spinful as although the horse is the council's, the whole of Cherhill Down is NT owned and they grant full public access so if someone asked for permission it wasn't merely to go for a walk.
Anyway, the notice was on their land, wasn't it?

Should we write and ask them if it's OK to put a "No Brandalism" notice up there? (For free, as it's a good cause).