1. Typical stuff that they try to simplify science and thus teach stuff wrong, like the classical each area of tongue has a specific taste receptor thing.
2. Government heavily interfere with what is taught in classrooms regarding some subjects, most notoriously the government is quite heavy handed in anti-monarchy rethoric, I was actually shocked when I was in my 25s and started to find out a lot of stuff I learned as a kid were not just misrepresentations or wrong interpretation of things, but outright lies and propaganda, one interesting example: schooling here blame a lot of slavery on monarchy and nobility in general, meanwhile the monarchs actually wrote anti-slavery essays, and even made a deal with England at the time authorizing them to attack slaver ships, at some point the government even made it clearer, told England to treat all slavers as pirates, and gave them permission to even attack docked slaver ships freely. Also schools here teach government was "absolutist", but it wasn't, only reason monarch was letting england screw with ships, is because whenever he tried to make a law ending slavery, with nobility support, the liberals would vote AGAINST it (and mostly out of stupidity too, farms that stopped using slaves had bigger profits, maintaining the slaves was more expensive than paying the tiny wages they were paying to the immigrants).
3. Schooling here has a sort of mandatory political bias, not just as in... most teachers being left leaning (something kinda normal in most of western world), but as in the government explicity dictating that certain subjects need to base their theories on Antonio Gramsci, there is some infamous youtube videos where right-wing teachers argue against that with public officials, in one of the videos a teacher outright ask an official what about the parents rights to not have the kid be forced to learn that, and the official replies that the state knows best, that if they don't want their kids to be indoctrinated, they have to pull the kids from school, and face the consequences.
4. Schools here also teach a lot of useless stuff, with teachers insisting you will use all of it... but I guess this applies to most of the world.
But do you really think allowing homeschooling would improve any of this on a societal scale?
Sure, some people will do an excellent job at educating their kids, but many will indoctrinate them even more than the public school system ever would.
If a lot of parents indoctrinate the kids to be conservative religious.
Another group indoctrinate their kids to be far-left.
Another group is anti-science.
Another pro-science...
When all these people vote, you still end up with politicians that actually represent the society.
Now if everyone is mandatorily indoctrinated in schools that teach whatever the government wants, would that be a true democracy? What happens when actual science turns to be anti-government, do you think the government would remain pro-science?
That's why education has to try to be as objective and apolitical as possible, and when the two are at odds, choose objectivity. And accept that it's never going to be perfect. Nobody is ever going to agree on where that sweet spot is and there will always be segments of the population who vociferously disagree with how some things are taught, but as long as there's a critical mass of educators making good faith efforts to achieve those ideals, the result will be imperfect but better than kids being taught random things (whatever their parents want to indoctrinate them with) by random people who aren't professional educators (the parents doing the teaching). I'm not arguing against homeschooling here, just arguing against the nihilism expressed by the parent post. I'm fine with homeschooling as long as there are educational standards and it's not just parents who want to brainwash their kids.
FWIW I was partially educated in American public schools and got no shortage of what many would consider to be anti-government messaging, at least in the sense of being encouraged to think critically on my own and taught about all sorts of occasions where people's moral convictions went against government policies so they protested or went into government or whatever to change it for the better. I also got no shortage of focus on all of the ways in which America did terrible things in the past: Native American genocide, slavery, Japanese internment camps during WW2, Jim Crow laws, etc. The focus was always on why those things were bad, how they got changed, and how understanding those things helps us perceive normalized shittiness in the present day so we can fix it and make things better. Nothing's ever completely fixed, history is just an endless process of things (usually) getting marginally better over time in the aggregate.
I'm sure there are lots of shitty schools with shitty teachers that teach blind obedience and tribalism, but those are shitty schools with shitty teachers and they need funding and improvement, rather than throw our hands up in the air and say there are no standards and no truth and everyone just believe whatever. If nothing else, education at least needs to expose people to a range of things they wouldn't be exposed to if left to their own devices, and help students learn critical thinking so they can make their own rational choices about things.
Also it's not like parents give up all rights to educate their kids when they don't homeschool them. They can teach their kids all they want about whatever they want, including things that contradict what's taught in class, for better or worse.
