They can afford to send their kids to private school, they believe that school to be superior to the public schools their kids would otherwise attend, so they pay the private tuition. As a parent, I agree with this. I, however, do not have a $400k income with a $169k expense account and a sweet pension plan, like he does.
In short, his kids don't attend public school because he can afford for them to go somewhere better. I am not willing to penalize any child for the sake of their parents' political views (or offices), nor am I willing to restrict people from spending their own money just because they might buy something I don't like.
I could be mistaken, but as far as I am aware no politician (from either side of the aisle) sends their kids to a government school. At least, no politician at the national level (president, congress, etc.).
I am not a fan of having a centralized governmental school system (I think the indoctrination risk is greater, because then there aren't competing institutions around to teach anything different) but it strikes me that any voter who does believe in government schools should demand that their representatives send their own kids to governmental schools as a sign of good faith.
About half of Congress sends their kids to public school.
http://www.publicschoolreview.com/blog/how-many-politicians-...
In the matter of experience in dealing with the issues (and dealing with the Secret Service on security is far from the only one) that attend children of major public figures, no, a school is not a school. And Sidwell Friends, the school that has been attended by every school-age child of a US President since the Nixon Administration is, arguably, sui generis in that regard.
The President not sending his kids to public schools is probably, as much as anything, a service to the public school that they would go to if they weren't going to Sidwell Friends.