In what education system are outcomes guaranteed?

The story you linked to is pretty interesting, but she clearly states she was allowed to learn (just not forced to), she obviously understood what a university was, and had the desire and ability to apply and be accepted to university at an age when her parents would have had the legal right to stop her if they so wished - they obviously supported her application. Her outcome was clearly a lot better than many students who go to public school.

This story is fascinating (you really should read the whole book). But using her as example is completely misleading. She is a genius (I think). Very similar to the movie "Good Will Hunting". She got accepted into Cambridge with little to no education whatsoever. Even if you are a genius, this is not easy, since you just have to understand the system to some extend, no matter how smart you are, to be accepted into it.

Suffices to say, millions would fail under similar conditions, where she succeeded. Odds in normal education are much better than that.

I interpreted "the system" of your GP as wider society, not just the normal public education system, because of the first paragraph.

Also, yes Tara was given lots of autonomy early in her education, but she was completely unprepared for the world even at BYU (her undergraduate university), she struggled through many years of therapy, and she claims she was admitted to Cambridge because of luck (on top of lots of talent and hard work) because she still needed lots of remedial coursework to catch up. One interesting quote from a (TED?) talk she gave is that her early childhood was spent reading early Mormon scripture, so she spoke and wrote with the character of a mid-19th century LDS prophet when she arrived at university.

I went to public schools for 12 years and I still turned out weird. No guarantees either way.

I think the view that homeschooled kids are weird is caused less by homeschooling and more by the fact that weirdos are more likely to homeschool.

Let's say your kid is "weird" - on the autism spectrum, or ADD, or has severe allergies, or whatever. That can often cause problems in school, because children are often cruel. Parents of such children may be more likely to homeschool them than parents of kids that better fit the "normal" pattern.

Let's say the kid turns out a little weird. Would they have turned out better going through the public school system?

It's not necessarily fair to blame the homeschooling if the kids come out different. What were they at the start?