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Simply put you have to look long and hard to find a state school that fields a rugby team these days
Massively this ^
There are a limited number of minis setups that are out there (St Neots Minis near me and Battersea Ironsides being the only two non-school outfits I can recall at the moment). But rugby union introcduction and development for children is dominated by the public (and grammar) school system.
While I understand where you are coming from. I would add that the same thing historically applies to every home nation bar Wales. At times in the past the Ireland team has resembled a BlackRock College old boys team and if it wasn't for the borders the Scottish team would have until very recently been made up almost exclusively from ex public schoolboys.
Ireland especially has made great strides in moving away from this stereotype partly because, Munster, the other traditional rugby hotbed is vastly different from Dublin but also because the academy system is excellent and draws from a much wider community than is traditional.
I haven't read the article that @fizzy.bleach referenced yet but from my own experience the continued dominance of the public school system is down to the fact that clubs run the academies and prefer to deal with schools as it's easier for them. Simply put you have to look long and hard to find a state school that fields a rugby team these days.