Private education isn't necessarily the best thing: it depends on the child. What benefits do you believe a private school would bring that a good state school wouldn't? Your OP tends to suggest you believe this would open certain doors, which I'm interpreting to mean in terms of networking and contacts (correct me if I've got that wrong).
In which case, anecdotal experience (as a university lecturer) tells me that it likely won't. If you're not in those circles already, attenting really exclusive schools like Eton, Cheltenham or Marlborough, then attending public school isn't a golden key to buying your way in. (DH's cousin had a child at Eton. I can assure you it isn't always what it's cracked up to be - they've had all manner of problems with him).
We've opted for other ways to maximise our DC's educational benefits (just started secondary this year), and decided the local state school was the best choice. DC has dyspraxia (sensory processing disorder) and dyslexia, and is possibly autistic (assessment for that is coming up this term, which we are having to do privately owing to huge waiting lists). We've chosen to invest our money in the right diagnoses, our LEA being woefully underfunded, and specialised tuition targeted to help his individual needs. He attends weekly dyslexia classes and physical classes with a local Occupational Health clinic to help his balance and coordination.
Dyslexia was only diagnosed toward the end of Year 5 and he's been having these classes since Year 6. He received the highest possible score in SATS for reading and comprehension, plus 'greater depth' in SPAG and Maths - and was barely scraping 3-10 in his spelling before the tuition started. For a kid with SEN, which we've only known about for a year, I was blown away, not to mention very proud. He's now in the top sets for English and Maths (and I do believe very much in streaming as the most effective method). I also don't believe my bookish DC would thrive in the sort of sporty environment often embraced by the private sector.
Just an alternative viewpoint if that helps you, of educational decisions made very carefully and taking into account all our child's individual needs. We are also a self-made couple, and I do totally understand your desire to do the best you can for your child.