I didn't know home-schooling was so common in America. And I didn't know home-schooled kids were so successful in national tests compared to their peers, I would have thought they noticeably underperformed.

The problem is that, in the US, it does create those problems.

Most of our government schools are funded via very costly property taxes. If you own a house, you must pay >$1,000/yr in taxes which go towards the school district. If you fail to pay those taxes, the government can do some nasty things to you. Since the cost of education keeps rising in America, so too do property taxes.

Problem is, those 'Fancy pant private schools' cost less than government schools in most cases. For example, in Detroit, private schools are about $4,000/yr less than public schools, despite the fact that private schools are graduating ~95% of their African-American kids, while public schools in the D are graduating ~40% overall.

From my understanding, the reason private schools are so much cheaper is because of what the money is not spent on ... You're unlikely to see state of the art equipment in private school classrooms that sits around unused because teachers haven't been trained how to use it, the teachers make less money with fewer benefits and have substantially less job security, the schools do not have fancy fitness facilities, and there are far fewer people in administrative positions.

Even with all those "negatives" private schools tend to attract the best teachers, in a large part because of the consequences of these limitations. Since these schools can get rid of the worst teachers, and they attract far fewer of the teachers who are motivated by the benefits of being a teacher, working for a private school means you have to spend less of your time trying to fix the mistakes of bad teachers. Since money is not being spent on high-tech equipment that sits unused there is often money available for textbooks and classroom supplies. While there are fitness programs within many of these private schools, the more basic fitness facilities demonstrates their focus on academics; and the fewer administrative positions are a demonstration that the schools are far less of a bureaucratic mess.

Homeschool is for parents that want to brainwash their kids with their own brand of religion

Right ...

Because wanting to avoid the "high quality" education provided by the public school system must be driven by some wacky religious reasons.

Last I looked, private school kids do best, followed by homeschooled and then government school kids.  Government schools are the most expensive, followed by private (generally) and then homeschooled.

i don't see a problem with it.

At least in the US, some of the government schools also teach religious ideas.  I remember my middle school biology teacher explaining to me that evolution is just a theory that he thinks it wrong.  My high school taught the bible as a history class.

The majority of private schools in much of the US and the world are religious, but you usually don't have to actually follow that religion to attend them.  Sometimes, you just get a discount.  For example, there is a private religious school near me.  It's around $2,000 per year if you are of that religion (it is at a church and the church subsidies it's own people) or $3,000 if you don't.  Compare that to the area government schools which cost taxpayers around $17,000 per year per student.

I know some people that weren't homeschooled for religious reasons.  Heck, I live, um, very close to someone that is being homeschooled for other than religious reasons right now.  Anyway, the type of parent that is likely to try to force their religion on their child in the home is also likely to force them to attend a church, anyway.  And still, on average, that parent will do a better job educating the child in general than the government school system at perhaps 1/100th the cost to taxpayers.

America isn't the world, Americas system may be broken but that does not mean that the idea of public schooling is broken.