Not all private schools are like what you describe TPTP. Whilst i agree some might be like that, i dont think it is wise to over generalised.
Based on my experience, the private school my children attend (nursery level at the moment but they ll stay there fir primary) is very low key and parents there are very down to earth, the school is not in a fashionable area of wandsworth yet achieving outstanding results. More importantly the children there are nurtured and happy, and allowed to be children there. Which is the most important thing for me. "winning" and "achieving" come second to me. For us, Thomas doesnt fit the bill but i know Thomas is an excellent school.
The good thing about wandsworth is the luxury of choice being state or private. So pretty much anybody can find a school that suit them. Just the small matter of getting in...