Absolutely not! LOL I was the product of the state school system. Went to uni, worked in an office job etc.
It's really in your (or kids shall I say) hand's.
So how does it differ once in Uni? If both a private and state school kid get in. Or do state school kids have a tough time getting into Uni?
Getting into Uni is also not a gurantee you will make it. Not enough room for everyone to make it.