Not by much, I'd bet. If at all.
The poster seems to have confused top-tier private schools and gifted programs. Read enough politician and C-suite and such bios and it's very clear what's going on. You practically never see "attended a pretty decent public high school—but was in the gifted program!" Private college prep secondary schools (at the very least—often it's private schools all the way) on the other hand are overwhelmingly the norm in that set.
It's kinda depressing as a parent. If you haven't scraped together 25+k/yr for elite prep school tuition (and, probably, boarding) all your "you can be anything you want if you try really hard!" is kinda a lie. Like, that's still much better than not trying hard and will likely improve your life outcomes, but, looking at the actual world, realistically... nah, sorry, you're probably locked out of a lot of options. There are de facto requirements, and we couldn't afford them. Sorry kid.
Similar story with The Arts. You start looking at the backgrounds of very high-paid artists of all kinds (actors, musicians, even authors a lot of the time if they're considered good and not "merely" popular) and you're likely screwed if you weren't at least one of: 1) born to a family that's already successful at that, or 2) had an expensive and very focused education starting before college. Lots of the successful folks had both of those things. Again: there are counter examples, and it's technically possible to get in if your parents weren't in the arts and you didn't start gigging/acting/attending-an-artsy-private-school by the time you were 12, but realistically you're looking at a serious uphill battle.