You are generalising too much OP. These issues are very area-specific.
For instance, in grammar school areas, there will be snobbery about who got into the grammar and who did not.
For people who use the 7-13 prep school system, it's not about whether your DC is going to a private school or not (because they all will be), but a case of which one. Most will be going to boarding schools at 13. All sorts of shenanigans on that front. The preps in places like Berkshire are notorious for all this.
Where I live in SW London, we are surrounded by the likes of St Paul's Girls (and Boys), Godolphin and Latymer and Latymer Upper as the 'top tier' schools. Then, closely behind, are schools like Putney High, and a few others. Then there are the 'back up schools' (no these are not comprehensives in special measures, but independent schools that are still very selective and get very good results to match). So round here, it's not about going to a private school per se - because you might not even know anyone who uses the state secondaries. It's about which school they are able to get into via the exams and interviews. Some schools have 1,500 applicants for 100 plsces, si you can imagine the tiger parenting going on. Many families will send their DC to boarding schools as a way of saving face if they don't get into the very top tier schools - "oh but they di actually need space for their sporting genius to flourish, you see, and we could only find it at Marlborough. He simply isn't a fit for St Paul's, he's too well-rounded."
So to summarise, I'm not sure if people brag in other parts of the country about using some local private school, but in my area- specific experience, going to a 'back up' (less selective private school) is not seen as any great shakes, no.