Sure, government can fund things without providing whole services but even the education they do provide is competing with private schools; they don't outlaw those. No monopoly; which again, doesn't apply to government which is always a monopoly.
The next step people often forget to think about is what happens when you fund something with gov money. You have accountability (real or lip service) and regulations that are necessary. Most private schools would not like vouchers. When they have to treat everybody equally it will not be so great for them. Then you have the whole religion issue where they can't give money to fund religious indoctrination of children; which is one reason many people go private (in my state most the private schools under perform but the faithful have no problem believing their schools are superior... or that the lesser education is not as important than brainwashing their children into not thinking for themselves.) I've been to private school, BTW.
Charters are the current fad. They do not perform any better on average and cost a great deal more money-- this is despite their ability to chuck all the kids they don't like! You'd think they would average out better given that HUGE advantage they have over public accept-anybody school.
As far as the invisible hand of the false god; the market... that is BS. Wake up to reality. Consumers do not have much say or care much - the impact is there but it is not absolute. Look at how Americans screwed their own economy with the rise of Walmart and other corps who ruined everything - it doesn't take much indirection and the consumer will fuck themselves over eagerly. It also has the problems of a direct democracy where only a few people can be experts and nobody can keep up with all the issues going on so people couldn't run anything larger than a single person could run (and while holding another job, having a life etc.) I don't buy a huge list of things but I can't keep up with all the boycotts. Then you apply this to education where parents do not know jack shit about education and the majority doesn't even care enough to get involved like they should be doing (remember, most people have both parents working and more combined hours than in the past; the time constraints alone are a problem.)
So, that private school going to open early so the poor kids who DO NOT EAT at home can get breakfast? nah, they don't allow those people in the school. Voucher schools can add fees on top of the gov money to do away with that... unless you regulate them; then they take in the minimum amount which is likely not enough.
Educators in most states are required to take continuing education themselves. Depends on how the program is run how well that works. The private colleges cater to the teachers the best with lots of pure BS courses that let you off the hook. I know educators. I even taught a course for them which surprised some because it wasn't a BS course like they expected; they made a mistake of not going to a private college. Don't know how bright they were, you don't take a course on robotics and computers in education and think it will be a joke... Like those courses on multiculturalism where they just go around town eating ethnic food (I'm not kidding, that is a course! not at my less prestigious public university but the fancy private college down the street.)
As far as latest education research-- teachers are the worst at learning new things! They are extremely set in their ways. I think it has to be a result of conditioning; they spend decades doing similar things that work well enough for them so it should naturally be hard to get them to change. Even if your great new thing works well, it may not work well for the individual teacher or the subject matter or the demographics. Sure, fire them and get a new sucker who is into the new fads and you might not end up any better for years while they get broken into the job. Although they can have advantages, educational fads do come and go. Some have bad results long term, there is a history of failed programs that did HARM to the guinea pigs; err, children. So adopting something popular is not a good thing. Being traditional is safer. The actually reasonable approach which never would happen (at least in the USA) would be to take the children doing well and stick them into schools that are traditional. The children who do poorly need to be put into schools that try out new experiments on them. Lumping people together by random or baseless factors (like income) isn't a formula for success. Yes, in some cases this would likely be a huge problem... a school full of ADHD children would be a nightmare! Ok say you do this, then people would get upset their brat gets less funding because traditional school is cheaper and the ADHD school gets more money... Psychologists... learning issues are mostly emotional issues, but we fund cops and "CEO"s in the school over psychologists.
Measurements are the huge problem. huge huge huge problem. People couldn't pick out CPUs other than looking at Ghz... If you did find the magic statistics the consumers wouldn't understand them. The political football education has been made into is why it's going downhill. It would have been better off saying the same since the 1960s. There are so many factors to measure as well... Like fact vs opinion-- 40+% of college students entering today? (that was probably last decade) well, they didn't know the difference! I doubt that improved. Fact vs opinion was taught to me ..3rd grade? Perhaps they stopped... but I'm thinking they still do it but the culture is so bad that kids end up all wrong by adulthood. I didn't even get into cultural issues! Like I said, you don't fix things with a 1 size fits all solution and we mess with what works whenever we try to fix what does not. No, again lots of smaller private schools do not fix the real problems. Parents don't know what to choose-- they'll just end up at Walmart Academy because it has low prices and high test scores (but only produce good walmart workers/consumers.)
I'm well aware of public school issues in my state (one near the top until no child left behind) and it is not perfect. But the solution is not to simply attack unions or the disrespected and generally underpaid teachers. That doesn't solve anything. Even if that helped, it can only be a tiny aspect to all the issues involved. I think the only way there is any hope is to remove politics from the equation because modern US politics is completely toxic to anything it touches; that means voters not responding to the issue as well as not polling it as a priority.
You go on believing your 1 maybe 2 factor solution will solve X. As well as thinking the only reason X hasn't been solved is because your solution has not been tried (or if it has not, it wasn't tried hard enough.) The reality is we have no simple solution and the problem will ever never be solved to peoples relative judgement.