it is true that the president may have no special expertise in assessing whether private schools are better than government schools, but it strikes me that any support he expresses for public schools is going to ring hollow if he isn't personally invested in the success of government schools.

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.