Hell no. The U.S Congress can't even pass a budget every year. How the hell is it going to oversee nationalized education? And many public schools, especially in the inner cities, are a disgrace. I'm not a big supporter of the Private Sector, but I understand that the U.S Government hasn't been able to do a lot of things correctly and solve big problems in a long, long time. Hell, the Government(state & federal) doesn't have a damn clue what it's going to do about 12 million illegal immigrants. Then, you want to throw nationalized education on that? Hell no.
Now, should the Private Sector have complete control over education? Again, hell no. The private sector can't even figure out how to distribute the wealth it makes. If it can't put its workers first, how the hell is it going to put the nation's children first? And the private sector, specifically corporations(which would have a lot of influence in a nationalized education system), rarely thinks long term. So we'd have Corporations influencing public policy that leads to short term profit, for short term increases in stock prices, for short term satisfication of shareholders. Hell no.
I don't know how long this education system has been this way, but it doesn't seem to be in any other direction than in decline. Now, what can government do about this? Nothing. At least not in this system. Not with politicians. The masses are going to have to solve this one. But there doesn't seem to be much hope there.