Gatwick does have a lot of quiet periods, among the very busy ones.
It could easily take slow GA. They would need some sort of slot system, a bit like Friedrichshafen (EDNY) when visiting the AERO 2007 exhibition. One books slots on their website and - Germans being organised - you can buy them with Paypal for a few Euros.
The trouble is that nobody in the business is interested in doing something radically different, when they get £1000 or whatever from every jet. It's the can't-do culture that's the problem, and equally it's the American can-do culture that keeps things going over there.
If I was running an airport, I would sell landing fees on the web, payable by CC or Paypal. Why have pilots queing up to pay landing fees? It's a waste of their time.