I don't really see why handing over control to the mayor would be an improvement, however. Sure, it eliminates bickering between board members and the interest groups (parents, teachers' unions, and the government school bureaucracy), but without some sort of competitive pressure on the mayor's administration, why would he be able to do any better? All he can do is decisively favor one group, or no group--I see no reason why he would be better able to get things done, and I see no strong incentives for improvement.
Suppose we built our cars like we provide education. We would have local factories, run by the government and paid for with property taxes. What car you would get would depend on where you lived. If you didn't like the car you got, you could move to another area, or go to local car factory meetings (where you must battle the workers' union, which wants maximum wages for minimum effort, and does not want their quality scrutinized, and the government bureaucracy, which cares little about the outcome). You could also buy a privately produced car, but that would mean that you would forego all that money you paid into the property tax system, and you'd have to pay the full price of a private car on top of that. Does anyone think this system would give us better cars? Surely not. It provides choice of cars only for the wealthiest buyers; the rest must consume the cars produced by the local monopoly.
Education is a very complicated good (or if you prefer, a very complicated investment), but I don't think the comparison to cars is unfair. Cars are also complicated bundles of characteristics--fuel economy, acceleration, comfort, appearance, reliability, safety, convenience features--that must be bought together. Some characteristics, such as reliability, cannot be immediately observed. Organizations, such as Consumer Reports, exist which try to collect data on reliability, so that consumers can make informed judgements about the long-term wisdom of buying a particular car. Is education really so different? Surely it makes a great deal more sense to set up a system of incentives that encourages schools to compete to improve quality, in order to attract students, rather than debate which whether we should place the government monopoly under a board's control or a mayor's control. Sure, we might get lucky and elect a particularly motivated, competent school-board director, but what about when he is gone? A system that relies on the public electing brilliant, motivated officials, rather than relentless competition, does not inspire my confidence. A system of incentives is more powerful and reliable in the long-run than a system of "big men".
A voucher system is one such method; give each student a voucher good for some amount of money, and let the student (and his or her parents) spend it on whatever school the student wants. Schools must lure students, and their money, to their doors, by providing education that people want. A common objection to this system is that, because students could choose private schools, this removes resources from the public school system. This is true, but it is also true that this system removes students from the public school system; with fewer students, the public school system does not need as many resources. The evidence on voucher systems that have been tried in the U.S. so far is mixed; the experiments are sometimes short-lived (teachers' unions strongly oppose them), and the lag times involved from input to output may mean we haven't given them enough time. Other countries, particularly some European countries that do better than the U.S. on standardized test scores, use a voucher-like system. (There are other reasons why they may do better on standardized tests, including the possibility that they test fewer of their students.)
Even if vouchers are not a panacea, I nonetheless see no good reason why education must be produced by government schools. Why should we expect the government to be better at producing education than a private institution? If equity or access are concerns, vouchers address them. I can think of a bad reason why government would want to provide education: to indoctrinate. Looking back on my own education, I find it bizarre that I was required to stand up and salute the flag, and in elementary school, we even had to sing The Star Spangled Banner every morning.