I'm probably way too late for this comment to not get buried, but in my experience 99% of homeschooled or "unschooled" kids growing up in the US around me had parents who were obsessed with avoiding secular education on things in basic biology (evolution) or history (slavery), OR had parents in the 90th+ percentile for household income and/or focus on education and STEM stuff (e.g. one parent took a part-time job teaching them at home to give them way-above-average skills compared to their peers entering HS or College). In all cases, the "better" outcomes were situational and largely correlated with income and access to opportunity rather than a homeschooling model actually being proven to be better.

AFAIK, most academic outcomes for home-schooled kids are actually worse when adjusted for income. This is at least true for private school vs. public school (https://www.usnews.com/education/k12/articles/private-school...) but I suspect home school data is much more sparse