I would point out as others have done, that there are some fantastic Independnet schools which offer something. Quite different to state schools, there are lots of good independents and there are also lots of very third rate ones.
Be careful not to decide that private just for its own sake is something to be desired. It's only ever worth it if it offers something more than the state option and where fees are considerable and will impact your family, the difference or extra it offers needs to be significant too.
So I would return to the drawing board at each and every stage of education. Rather than just deciding you want private, research all the local options, both state and private and decide at each stage which is best for your child and pocket.
Re the working class thing....all I would say is that if you and your family have no prior experience of Independnet schools, you might decide private is better than state and see private as an aspirational. It's all about gathering information and making informed choices based on exactly what is available in your area. Many parents find it hard to compare education within the state system, never mind between state and Independnet as well. Educating yourself about what to look for and questions to ask is important.
Remember, Independnet schools will be able to put on an impressive display. Their facilities should be good and they might pressure you to choose quickly and pay deposits. You need to look beyond the shiny facilities and fluff to the actual education.
And remember, many who can afford to pay, choose not to. They carry out the research mentioned above and conclude that for them in their area, the fees aren't warranted, because the state education available is perfectly good.
Private isn't always better and it's not always about class either. More and more of the middle classes are choosing not to pay, partly because some can't afford it, but also because many state schools have improved a lot.