While I hate the solution. There are many problems with it.
1) Why do parents have the sole privilege in deciding where the money goes? It's my tax dollars. In the current system, I can go to school board meetings and try to change policy, even if I don't have kids going to school right now.
(Of the many possible scenarios, I might want to change things because I know my kids will be going to school in a few years, or based on experiences from having kids who just graduated.)
2) You write "some bar set by the government." Who sets that bar? Is it the local area, in which case how does it differ from a school board? Is it the entire state, in which case, how do local people affect what's going on in their school?
Or are their multiple organizations which can be authorized to provide oversight, in which case, can't the private schools pick the one with the least oversight? We see that already with some charter school systems.
3) Since these are private schools, can they decide to not accept someone? If so, based on which criteria? Can a private school decide that it's too expensive to school someone with emotional problems (a "disruption"), and expel that student?
Public schools are required to accept everyone, with a high bar for expulsion. What happens if private schools only accept the children which are cheapest and easiest to educate, leaving the more difficult children to the public school system?
What happens if no private schools are willing to accept a student, and there no more public school system which is required to accept everyone?
4) You mention that it's "competitive in how they produce students".
In many districts there is only one school. There may have been more schools before, but they merged in order to save costs, because it's cheaper to have one large building (and bus kids in) than many small building, each with its own staff.
Surely having several private school systems has its own overhead.
How do you know there there really enough economic savings in this competition to be able to overcome the economy-of-scale savings? Especially in districts where right now there is only one school?
Plus, the large public high school I went to offered a wide selection of courses. There was enough interest in Latin, for example, to have a Latin teacher. Break the school population into three, and no one school will be able to afford a Latin teacher. Sure, one might, but then that school might not have a jazz band class. A student who wants both Latin and jazz is unlikely to get that if there are many small schools instead of one large one.
This suggests there may be a natural monopoly, rather that competition, for many districts.
5) You assumed "that the primary reason to choose one over another is test scores".
Test scores are highly correlated with family income. If we choose schools based on test scores, then we send students to schools in rich neighborhoods.
Test scores are pretty worthless in the first place. Test prep is effective, but takes time away from all of the other topics (art, P.E., home economics, music, foreign language, shop) which are also important - if your goal is to produce educated citizens - but aren't tested.
6) Most good private schools are more expensive than good public schools. That check will end up subsidizing rich people to send their kids to expensive private schools. It won't poor people send their kids to good private schools.
7) How much of the school budget will be spent on marketing in the system you envision? In a public school system, there is very little marketing overhead.