You don't want competition. The only competition a school can have is by proxy measurements, and then you are deep into that age old problem that what you measure will decide the outcome.
Just like paying programmers for each bug they fix, which will drive the production of bugs to fix, paying schools for grading children will earn you a lot of highly graded children.
What we need is a respect for the profession. Treat good teachers with a lot of social respect, pay them well so they don't need to think about anything else, and give them a lot of opportunity to educate children as best they see fit. Just like we do with the best doctors.
Just like paying programmers for each bug they fix, which will drive the production of bugs to fix, paying schools for grading children will earn you a lot of highly graded children.
What we need is a respect for the profession. Treat good teachers with a lot of social respect, pay them well so they don't need to think about anything else, and give them a lot of opportunity to educate children as best they see fit. Just like we do with the best doctors.