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All three options can be fantastic, and all three options can be terrible. Generally (speaking for the USA) there is a correlation between quality and cost in each option. To quote Supertramp, "they say it all depends on your money and who is in your family tree," and there's something to that.
Public - houses in the best school districts generally cost more to buy and/or have higher property taxes (in gross amount if not rate) than houses in lousy school districts. Not everyone can afford to live in them.
Private - the really excellent private schools generally have eye-watering tuition. Not everyone can afford to pay it.
Home School - The opportunity cost of having one parent take time away from generating income can be significant, and presumably doing it well takes even more time than doing it poorly. Not everyone can afford that (and not every family even has two parents if one has died or taken off). Not to mention you need to be lucky enough to have a parent in the home with the talent and temperament to pull it off.
Two things I've observed in addition to these generalities -
The best private High Schools can definitely confer an advantage (beyond what they deserve on just the merits of the education) over all other options if your goal is for the student to gain admission to one of the two or three dozen most highly selective colleges/universities.
In public schools, even if you are in excellent district that has overcome all the normal challenges to achieving that status, you have to be alert (these days anyway) for stealth or not-so-stealth attempts to take over the school board by folks with a variety of fringe political agendas, usually in some combination of anti-vax and/or book banning (or removing everything in the curriculum to which even one parent objects) and/or trying to dumb down the curriculum with "PragerU" type nonsense. It can be frustrating. We've had some districts nearby go through some pretty serious back and forth battles on all that stuff, which is a real shame and a real distraction. Luckily the district in which I live, while it has seen such candidates, hasn't voted any of them in.
I don't know how anyone would be able to make a blanket statement that any one of the three options is best for every family regardless of their circumstances.
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