If OP's kid is currently in primary then the whole argument over A levels is a bit redundant.
Having experienced both state and (a selective academic) private during my primary years, my perspective is that primary private education is largely a waste of money unless your kid has SEND, has struggled with a state in the past (e.g. bullying or highly introverted and struggled with class sizes), or unless you have issues with wrap around care (e.g. no easy options at local state).
The real difference was at the senior school - smaller class sizes, more personal pastoral care, high quality extra curricular activities (leading facilities for rowing, swimming, athletics, climbing, hockey, great music department with successful alumni, drama department with lots of famous alumni, great art department and a private school-owned art gallery in town which regularly showcased UK artists alongside artwork from pupils), and as it was an academic selective school no-one was bullied for being nerdy/a swot - working hard was the standard and other students would encourage one other to study.
The school had a lot of academic SEND kids that were bright but had struggled in mainstream state schools and the school was also good at identifying undiagnosed kids to refer for SEND too which had lifelong benefits to those diagnosed early.
There are, of course, private schools that don't have the same facilities and ethos. However, I think it's shortsighted to throw all private schools into the same bucket and assume A levels are the ultimate measure of success and comparison between schools. I know kids who got top marks and have really struggled with their professional lives, and I know many kids who got mediocre A levels and thrived. The difference between those who thrived and those who didn't was largely due to confidence, tenacity, family support, and having some level of humility and charisma.
People with contacts willing to engage in nepotism will be looked after regardless of school background or A level results, I'm talking about the general public when I comment on the above.
As for my choice for my own child, it'll depend on how she fares in the local (excellent) state school and our family finances. I agree that introducing a stressful financial situation in the household for the sake of sending kids private doesn't necessarily balance out the benefits of private schooling. But if my kid is truly unhappy in the available state schools, then I'd be willing to explore how to make finances work to make private school a viable option for her. This wouldn't be motivated by exam results - purely for ensuring her confidence isn't crushed by a system that is unable to support her.