As uncomfortable as it is, the reality is that unless your parents are part of the establishment before you even step foot in the door, you are unlikely to benefit much from networking opportunities or door opening.
People who consider themselves elite (emphasis on consider) can smell middle class opportunism a mile off. They'll support their own, but not the children of middle class professionals who have little to offer them.
In my experience children tend to form groups strongly tied to socioeconomic and social status from secondary school onwards. In my experience the Camillas, Fenellas and Prunellas at my school had no interest in friendship with the children of middle class professionals.
Without the networking benefits and door opening, you are left with smaller class sizes and better facilities. Great for some children, particularly those with SEN, or if someone prioritises private schooling above all else and understands the financial impact. A bright child living in an area with a good state school is likely to achieve their potential without the need for private education and attending independent school can set them and their family back financially. Private school may be a dream missold to the middle classes, who (as in this case) may struggle to afford it and often cannot realise its true benefits.
As middle class professionals, we opt out and our children will have access to rowing, shooting, opera, music, gym & sports lessons as these are our hobbies/interests. We can give them whatever holidays and enrichment activities we can offer, but it won't be costing them hundreds of thousands of pounds in opportunity cost and we won't work ourselves into an early grave to pay for their education. We live in a smaller house than we would like in an area with a good state school so we can get our children into our local secondary (appreciate that is not an option open to all). By not paying for private school, we can spend more time with our children, and that is what they will remember when they are grown.
Middle class parents who can barely afford fees can make whatever decision is right for them and their family but ignorance of the financial ramifications/social dynamics of the system could ultimately stop their sacrifice feeling worthwhile.