Yes, this would definitely be an improvement but would not fix the problems that are intrinsic to government owned and run, single-payer, and socialist entitlement schools( even if managed by district as small as a city block or suburban housing division. )
Why? ( I hope you will thoughtfully follow my reasoning.)
Answer: Fundamentally, government owned and run schooling is a First Amendment and freedom of conscience abuse regardless of how small the locally run district might be.
1) No school can be religiously, politically, or culturally neutral in content or consequences. Such a state of neutrality can not exist in the mind of any sentient being, nor can it exist in a school. Even if the district was as small as a suburban housing division, a more powerful group of voters would be imposing ( by police threat) their NON-neutral religious, political, and cultural worldview on a minority faction and forcing them to pay for it and use it if they can not afford or find an alternative.
2) Government owned and run socialist entitlement and single-payer schooling is risky for the child. As the child grows older it is impossible for him not to notice that the voting mob is giving him tuition-free schooling! Gee! If the voting mob can force others to pay for schooling, why not use that mob to get **lots** of “free” stuff? Simply attending these socialist-entitlement schools may teach the child to be comfortable with socialism and government force to pay for and use the socialist service. This is true even if the district were the size of a city block.
3) Compulsory schooling risks teaching the child to comfortable and compliant with government compulsion. ( Do you think our Founding Fathers would have tolerated a TSA search for even one nanosecond?) Even if districts were a tiny as a small town government schooling teaches government compulsion.
4) All schools must strictly suppress First Amendment Rights. When government does this in its socialist-entitlement schools, the child risks learning to be comfortable with government suppression of God-given human rights. This would be true even in a tiny tiny district.
Solution: Begin the process of privatizing education. The most ideal goal would be to have complete separation of school and state.