I've discussed this before in other posts. I used to manage School Sport Partnerships and we had an upsurge of coaches going into state schools, which meant more playing and much more competition (contrary to reports!).
We were working with local clubs, the local CCC (in my case Northants), who in a lot of cases hosted school competitions on their outfield. We had strong, sustained links with the pro clubs and the ECB, who were providing coaches as part of a strategic coaching plan in state schools.
Unfortunately, the funding got cut, many of us managers were made redundant and this topic is getting aired three years too late! Only now are many seeing the impact of the loss of a specialised state school sport system that saw participation rise for 2+ hours per week from 23% in 2002 to 94% in 2010...
With that in mind, I would suggest that public schools keep doing what they're doing - i.e. feeding the elite game. What needs to happen on a local level is more support for clubs and the generation and sustainment of school-club links...