It’s more about the whole package. I homeschooled. Thanks to COVID and some ensuing frustrations with the district, my kid has done public, private and homeschool at different times. I’m in a moderately large metropolitan area, for context.
If a kid has a good environment at home — safe, fed, loved, healthy, encouraged and given access to do things they’re interested in — they’ll do great. Public schools are mostly in the business of serving kids who have problems with one or more of these things, so if you can provide all of them you are not their target audience. Such families don’t have the problems that the district spends most of its energy thinking about.
IMHO, that’s a situation where homeschooling shines if the kid is ok being alone a lot and also has a good social life. (Young kids tend to play with neighbors anyway; older kids move in friend groups, so the game is to get them opportunities to meet other kids until they can break into one of those.)
A lot of people get stuck on “can you really teach everything yourself?” No. You can’t. But that’s ok, because it’s a completely different process to school and the skills and resources you and your kid need are different. Your kid will need to self-teach more and more as he or she gets older, and your job is to make sure they have the resources and encouragement to do that. If they’re not independently interested in doing that, it’s probably not a good fit.
Alternatively, you can pick a private school. These are costly to families, which can make it seem like they’re rich when in fact
they often have less funding per student than a public school. What they DO have is a more tightly focused mission. Private schools also tend to target families who are able to provide that strong home environment, and who don’t have serious learning disabilities or behavioral problems. Thus, they don’t need as high a budget per student, as they can skip many of the most difficult and costly responsibilities a school has. Their governance structures can vary, but in general I think the fact that they’re outside of the normal election process helps them define a more coherent set of principles and consistently apply that over many years. They’re also smaller and so information has fewer layers of bureaucracy to penetrate, and decisions can be made faster. This shows in everything from curriculum and facilities to discipline and staff morale. Some private schools do this better or more nobly than others, but public schools struggle to do it at all due to the realities of electoral politics. Thus, the private schools near me have tended to be no-phones-allowed for many years; the public schools are only now and with great effort able to implement that, and even I think it will be hard to make it stick.
Most importantly… it’s about what the kid wants. Do your best to avoid making your kid spend his or her childhood somewhere they don’t want to be, doing stuff they don’t value.