> Which is essentially what public school does.
The school I went to had, luckily, excellent teachers. One of them, not sure if as a coordinated effort or not, was big on letting the class decide things and helping us form the social structures needed for that - structuring discussions, votes, rules, and so on. I suspect it was a reaction to the dictatorship time requirement of studying an idealized version of Brazil's political organization.
A first step is to properly fund public schools. Then one would need to better select teachers (which becomes an option if teaching pays better) and train them. Teachers need to be trained in teaching, not only their subject matters, and need to be kept updated.
There are a lot of schools that get funded quite well and perform very poorly. It matters how you spend that money. The data on this is hard to properly interpret too since higher income parents tend to live in higher income areas which means better funded schools and those parents also tend to be more involved in their childrens academic and social success than lower income so the correlated variables get confusing.
However I do agree that spending funds in ways that help a public school compete with private and/or homeschooling is a worthy goal. That means you need:
1. more teachers per student in the school. (More admin doesn't solve this. You need smaller class sizes). You'll never be able to afford the one-on-one ratio a homeschool family can achieve but you can certainly close the gap.
2. more focus on actually proven approaches to the basics of reading, writing, and arithmetic. (phonics, essay writing, literature analysis, and "old math") A lot of pseudo science has entered the class room over that last few decades.
3. fostering more parental involvement in their child's education. This is extremely difficult to improve but has the highest impact. One of the reasons homeschooling on average does better is because the parents are self selected for involvement.
4. holding kids to real standards and having consequences for not meeting those standards. Tricky to do politically but essential for kids to learn both social skills and academic skills.
> which means better funded schools
This is another problem - school quality influences real estate valuation. If I support my local school and it gets better results, not only I can claim tax credits for donating to the school, but I also increased the value of my house. Or a house near a school I support.
Donations to support schools should always be equally divided among all schools, period, so that these donations can't create favored and disfavored regions. That public schooling should be both universal and uniform should be a given.
There are definitely hero teachers and administrations in public schools. They aren't the norm though and it's a bit of a lottery whether your child will end up with one.