Before I retired I was a junior varsity coach, and to a great extent, your premise has great validity. The golfers on the varsity team usually were also members at local clubs. However, as the JV coach I would have about half country clubs and half municipal players.
I particularly loved this arrangement, because I had an opportunity to do more teaching than the varsity coach, who players usually took lessons from their local pro. To be truthful, not many of the muni players became starters, but occasionally one of the non-country club players that I worked with on the JV would become a starter.
One other note: programs like First Tee can make a difference in bringing golf to youth who normally do not have access to the courses or equipment.
Post script: Our varsity team has been first or second in their section of play for the last 20+ years, basically because the teams they play against have a lot more municipal player then we do.
I happen to belong to a private country club and I can tell you that some of the members' kids/grandkids have a significant advantage over the blue collar's kids/grandkids when it comes to playing at a high level. While excellent programs like First Tee are great they just cannot replace years of private lessons, first rate and fitted clubs and constant participation in youth tournaments. I see the kids at the club that are 10-12 years old with swings like the pros and shooting close to par on championship courses. So while I think golf is a game for the masses there isn't really a question in my mind that it is an distinct advantage for the kids from wealthy families to "make the local HS team".
Just so I don't come across as some sort of country club snob here I'll tell you I grew up in much different circumstances than I now find myself. So whatever I now have I got the old fashioned way, I earned it. But I can see from my club membership that there is a reason why wealth doesn't seem to last beyond the second or third generation. While having wealthy grandparents might make an easier road to being a scratch golfer some of these kids are (not all but many) in for a rude awaking when the start life outside of the CC.