That establishment is actually the pinnacle of their 'facilities'. Have you not seen where their 'academy' train, behind the Crystal Palace national sports centre? That truly is shameful.
Sure, agree entirely. the Community Pitch ought to be exempt, and I'd imagine that in reality they probably wouldn't care. Much easier though to have a blanket ban in place though, to empower the staff to prevent any photography of any importance.
Don't know. My guess would be that there is obviously some stuff going on that they wish to keep private, and that it is much easier to have a blanket ban on all public photography on site, than to task the the security staff with exercising discretion on what is, and is not allowable.