Mogha, thank you for those thoughtful comments and questions. To be clear, I was not advocating that education should be privatized, nor that the state should wash its hands from education. I was saying that the way the state intervenes in education should be aligned with the rationale for public intervention which, in this case, is one of externality rather than education being non-rival and non-excludable (I.e. A public good). And the reason I am concerned is that poor people are the ones losing out on the current education system. Nevertheless, let me address some of your points.
1. No one said teachers were lazy. The absenteeism rate is a fact. I agree that the reason is that the education system is dysfunctional and in need of reforms. That said, the evidence points to the fact that it is not low salaries per se that are driving teacher absenteeism, but the way in which teachers are paid and held accountable. For instance, when teachers are paid on contract (so that their pay is dependent on performance), absenteeism is less and learning outcomes higher (controlling for teacher qualifications, etc.). We also find that, on average, teachers in private schools are paid less and absenteeism is lower.
2. Your point about parents' in poor areas not having a choice is precisely why I was advocating that these markets need to be regulated. There is asymmetric information (parents don't know the quality of the school but the school does). We find that publishing information about schools' performance (in an accessible manner) can enforce some improvement in school quality.
3. The point about private lessons was an illustration of the clear demand for quality education in the face of a free public education system. Your illustration of students skipping school to attend the private lessons only reinforces the point. These private lessons are also hugely regressive.
4. I agree that the free public university is often of the highest quality. But this is precisely the reason why rich people send their children to prestigious secondary schools and succeed in getting them admitted to these universities. The share of students at these universities from the richest quintile is usually around 80 percent.
5. Because education is largely a private good, people are sending their children to schools and making sure they can read and write. That is why poor parents send their kids to fee-paying private schools, even if there is a free public school in the area. Society will get its engineers and doctors because students have an incentive to study and become doctors and engineers; we need to make sure the education system provides them with that education.