In East Harlem, they have more competition. Still, there's always the chance that your school could be closed if you don't excel. The fundamental reason schools don't change rapidly is that they have the next best thing to a monopoly. The only way, in a traditional public-school system, that you can have your children go elsewhere than where they're assigned is by paying a lot of money. Even when you take your kids out, the school district still gets your tax money, so it doesn't hurt them a bit.
Public choice is far superior to vouchers.
The problem under most voucher proposals is that parents could add money to the voucher. They could buy the most expensive education they wanted. Let's say a voucher was worth five thousand dollars a child. Some affluent people would add ten thousand dollars and send their kids to the fanciest private school. You'd lose the mixing by social class.