My answer would be that it sounds nice EXCEPT -
People would either have to carry their entire lunches around with them all day or rent lockers and then people would complain about that
People would want to bring in alcohol and that's not allowed
People would want to bring in glass containers and that's not allowed
People would trash the picnic areas - we've seen way too many posts about the state of the bathrooms which are trashed because people can't seem to put toilet paper and paper towels where they belong. I can't imagine a place with sandwich wrappings and juice boxes. Not a pretty site in the middle of a park.
It sounds nice and all but I really don't see it happening. You can currently bring in small lunches and enjoy them around the parks while sitting on benches or similar areas (not at counter service seating areas). Why would Disney change that practice?
Simply put - give people an inch and they will take a mile so it's better that Disney doesn't do this.