They also make the claim that charter schools "bleed off" the best students, which lowers the performance of the public schools.
Think about that one for a minute. Whether it is better for the individual students at charter schools is not the consideration. They are only concerned that the better students go elsewhere and therefore the average performance at their schools drops.
You are correct. Poorly performing public schools have a litany of excuses for their failure. They complain that parents aren't involved, but when parents get involved and demand better results the administrators and the teachers unions do everything possible to stop the efforts of the activist parents.
In the case of charter schools, traditional public school administrators and the NEA act against the most involved parents - those that decide to move their children to a school which fits the unique needs of the child.