Come on, you really don't believe that, do you? I doubt it, as even you know plenty of people who have succeeded in life that went to public school. I know I do, in fact I'm married to such a person.
I believe that it's an uphill fight and the hill is becoming much steeper.
Over the past decade or so I've witnessed a real trend toward quantifying employees by their educational background. This includes top law firms, banks, and even television. The kid who graduates at the top of his class from a state school is knocked out by the kid who graduated at the bottom of his class at an Ivy. This wasn't always the case, but it's trending that way more every single day.