At the limit case (and withdrawing from your 401k to support your 6 children going to private school is definitely a limit case), you can absolutely reason about what 'makes sense' and this simply is not it.
Sane Western countries have free and decent public education; and parents can't withdraw their future retirement early to fund dreams.
Your kid can go to an inner city public school, complete with going through a metal detector and x-ray every day, and go to Harvard. It's not likely, but it wasn't likely in the other school either.
You've misunderstood the cause/effect here. The kinds of people who go to Harvard (i.e., rich and/or smart young people) are indeed more likely to make more money and be elected president. But Harvard itself isn't what made it happen. It was being rich and/or smart.