This is false, choices exist. The choices may not be ideal or possible, but they do exist.
They also don't have a choice in where they live...it's either a trailer or no home. They don't have a choice on food...it's either unhealthy food or no food.
The problem isn't the educational system, but rather the income status. So instead of trying to change a symptom, change the cause.
Which is their choice. They do not have to.
How would that change? If there are three public schools in a city and one private school, how would having the option to attend X school or Y school change the fact they are receiving services from the same school district? And if they all choose to attend the private school, how is the private school going to be able to support all those students (books, technology, classrooms, teachers, etc.), when they do not receive financial support from the government?
The people who keep advocating this never bother to think this through. How would school vouchers help? Let's say I'm a troublemaking student from a poor family. Do you think I'm going to willingly go to a "bad" school? School vouchers would not change a single thing about the educational system, except enhance the shift towards educating those with money and leaving behind those without it.
Schools don't CHOOSE to be bad. Most schools will do the best with what they have. Some schools have much better financial resources than others. Some schools exist in better neighborhoods than others. Some schools have a student population which place a lower value on education than others. These factors determine quality of education. The idea that "competition" will improve schools is false, as long as school attendance is compulsory. All you would be doing is adding to the problems.