It is not private education, but public education that is the problem. The Federal Department of Education creates top-down mandates that take power away from the teachers and local school districts. Private education is one of the few places that sill allows for innovation and creativity. A voucher system should be adopted.
No, the private education system should not be abolished, because it helps create a competition in the public education system. For education, the more challenge that exist for schools to try harder for results, the better. With less competition between private and public education, there would be one standard across the board and it could be set to low with no incentive to challenge the students to go above and beyond.
No, I do not think that it would be a good idea to completely abolish the private education system. There is a lot of private schools that are very good schools, and teach their children just as good as any public school can, so why should we shut them down.
Everyone has the right to an education and if a family wants their child to be geared towards a specific type of education like religion then it is the family's choice. If school's were all public then those practicing religion would loose their control on implementing wanted religion outside their household. Private education provides a balance for the country.
Although the public education system is not perfect, it should not be abolished. Everyone should have a right to a free education in a free country. It would not be fair to make everyone have to pay for private schooling. It is better to make improvements in the current system rather than starting again from nothing.