My own education was in private, parochial schools through my undergraduate engineering degree, and it was outstanding. I went on to add a masters in business from one of the best biz schools in America, and feel very strongly and positively about the quality of what I received. Now, our two daughters-in-law are home schooling their children, and one of my wife's cousins home schooled both her now adult children. My own not very humble and bigoted opinion is that:
1.) It depends on why you want to do this. Does your child have special needs, e.g., do they suffer from ADD or social anxiety? Or are they intellectually gifted and you don't feel they will be challenged by the public system? Do you have strong religious beliefs that you want to be sure are taught? Or ?. If you don't have strong reasons why an outside school won't meet your child's needs, and all you're going to do is
to buy and administer a commercial curriculum, I'd ask why you want to do this.
2.) If you do it, *do it right*. Home schooling is much like working from home; it's very, very difficult to maintain any sense of structure and balance in your home classroom with all the other things going on in your and your child's life. One advantage to a public or private school is to remove them from screen time while the laundry gets done, etc. My wife's cousin used a specific room in their home as the classroom, and that communicated the seriousness with which she approached the process. It helped maintain the discipline and structure that are needed.
3.) I had a long career working in customer facing jobs and management for a major American electronics company, and hired and fired people. In a second, retirement career, I've taught math in a number of career schools in the last few years, and I have been appalled at the poor skills in basic arithmetic that I've encountered. Modern careers requires that one be able to both analyze and communicate. At the grade school level, this translates into the need to drill the students *every day* in basic arithmetic facts for yes, two, three, or more years. But many elementary public and private school teachers find this less rewarding than promoting reading and language skills. The children need *both*, but they need the basic arithmetic skills established strongly before they can move on to the creative stuff like algebra and trig. This will mean confronting, cajoling, encouraging, demanding, teasing, laughing,...it's a lot of work and it's intense, and I question whether most parents are up to that. To be fair, many public and private school teachers aren't up to it either, and evaluating your local schools is part of this decision.
4.) An intermediate solution might be to supplement outside school with material tailored to your child's interests. My very technically gifted father had me wiring industrial relay logic from blue prints when I was twelve, and that interest led me to engineering school. One of our granddaughters is visually artistically gifted, and we try to encourage that. Another shows signs of ADD, great physical energy, strong creativity, exceptional skill with numbers, and an insatiable demand for attention. She needs enormous one-on-one time, and until she gets a bit older, could be a real outside school classroom disruption.
5.) My wife taught for 23 years and has a validation in gifted education. One of her observations is that children have to be pushed hard enough to feel challenged, but not so hard as to feel unsuccessful. If your child is outside the normal range, either by talent or lack thereof, or by personality or strong unusual interests, they need to be pushed just hard enough in just the right areas. Maybe you can do a better job of
this, and maybe you can't. Be honest with yourself.
6.) You haven't mentioned religious beliefs, but if you're comfortable with a religious perspective, many parochial schools can be a better choice. Or maybe not; as I noted above, I got a killer education from parochial schools, but my own daughters ended up happier and more successful in public schools. We currently live in a city with a very large Catholic school system, roughly half the size of the public one, and many, many minority parents jump on any chance they get to put their children into that alternate system.
In summary: lots of choices. Make yours carefully and honestly.