Warren Buffett, for one.

Hmm ... Warren Buffet's a smart guy ... so what are the reasons behind it?

His reasoning is that if children of affluent and influential people are forced to go public schools, the standards for public schools would be greatly raised and there wouldn't be the need for private schools in the first place.

I'd go the opposite way: all private schools.  The government can provide a voucher for a certain amount of money (determined by the local school board) granted for the education of minors.  Parents can pick the school.

Either could work, both have problems.

1) In the no private school approach you've of course got the problem that the government controls all education of children.... and a slightly more "insidious" problem.

School funding.  For example, in my old state of Ohio.  Schools were givin funding largely by income taxes.  So high income tax areas got more tax money.

Some poor areas like cleveland got extra state funding that actually put it over the suburbs, but this is just to state how the fates of rich and poor aren't linked.

2) In the all private school approah, one could argue that this would leave out and screw small population towns and areas where there aren't bigger schools within a reasonable distance.

The real problem though isn't US schools.  It's US culture.

We have a two different kinds of parents.  Ones who boss their kids around a lot, have them do all sorts of extracurriculars, hit all the books on the summer reading list, go to museuems, read to them in their spare time....

and another group who lets there children have fun, see's schooling as the schools job primarily and that their main role in their childs education is to make sure they do their homework.

Group A is largely centered around rich people...

Group B is largely centered around poor people... 

The A style of parenting is much more effective.... lots of kids who are rich get the B treatment and they fail, other poor students get the A treatment and succeed.  The real challenge in education is getting children to work more year round.

If you check out the book "Outliers" they refrence a study where they studied the data on aptitude tests taken at the beggining of the school year, and again at the end of the school year to see how far a student has progressed.

When this was used to compare students by income level instead of to each other... they found a shocking conlcusion.

The gap between the rich and poor in apitutde was larger in the beggining of the school year rather then the end of the school year.

The gap kept getting bigger year by year.... but that gap was almost exclusivly growing during the SUMMERS and not the actual school year.

Summer vacation was the real problem for poor children.  Most other countries don't have such an extensive break.  In reality Summer Vacation is the real enemy.