Private school puts a lot of education in other people's hands, but you can choose a school which holds your values and works to inculcate them in your children.
Public school is free, but you have little control over value education, and the overall quality can be pretty crappy. You can improve this by living in a less desirable home in an area that has good schools for the same price as a nicer home in an area with bad schools. (Housing prices tend to following public school quality very closely.) You can also make up for it by spending time with your kids, filling in the gaps in their education and instilling values you consider important, or you can farm this work out part-time through youth organizations and lessons from private providers at some expense, but probably less than a fulltime private school.
Home schooling gives you complete control over education, but requires an immense amount of time and the quality is capped by your own teaching skills unless you farm out some of the education as mentioned for the public school.
My family selected the public model with a heavy emphasis on adding value through our own time, with some investment in private education, mainly for music lessons.
The values we worked to instill were self-reliance, compassion, practicality, and stability. We added to the quality of public education through encouraging reading, asking questions, and keeping a close watch on the educational quality of the schools and teachers and helping out whenever we could.
Instead of signing our kids up for a lot of "enriching" classes and activities based on what we thought they should do, we encouraged them to find their own passions, and supported those passions financially only as needed. For example, we paid for music lessons for our kids interested in music, because neither my wife or I are musically skilled. When my oldest became interested in movie-making in middle school, we provided him a used video camera, blank tapes (this was back in the early 2000s), cheap editing software, and technical advice as needed.
When they wanted to learn something we could teach, like cooking or answers to general questions, we made sure we were available to do those things.
All-in-all, we tried to keep it low pressure and self-guided, aside from some clear boundaries that evolved as they grew older and more responsible, and some expectations that we monitored and corrected when needed, including that they would do their best in school, and that they would behave at all times like civilized human beings who raise the overall happiness of the world by being here instead of reducing it. (For example, not tormenting siblings, acting properly according to the context, such as in a restaurant or public gathering, being polite and treating others with respect, not breaking things, especially things that don't belong to them, etc.)
Some of the results are already in. My oldest spent over three years in a youth gap-year volunteer service organization after high school, first as a participant, then as an intern, then as an employee. Now he is completely self-sufficient in his first year as a full-time student at a community college, with the intent of transferring to a four-year university and getting a credential to teach history to high school students. He sings and plays in a band at a church, and is engaged to a very nice young lady he has known since middle school.
My middle son went straight to the University of Chicago after high school, where he is pursuing an economics major, possibly with a minor in physics. Almost all the expenses (including housing and meals) are being paid through scholarships and grants through the University. We are kicking in a small amount (several thousand a year, mainly for health insurance), and he is paying about $4000 a year + books and incidental expenses through a combination of loans and part time work. He has a 3.8 GPA so far.
My daughter is a sophomore in high school and she is mainly interested in art, so besides the normal high school classes, she is taking an art class at school and participating in the art club, along with drawing at home.
We have been lucky to have healthy, reasonably intelligent kids. Our main expenses for them have been food, routine medical care, the cost of a larger living space, extra gas for taking them places, and occasional fun stuff like going to a theme park or movie (which we enjoyed too!). It was all stuff we could achieve on an average American family income, including some periods of unemployment.
I think when deciding whether to invest time or money, time is better for the kids, more fun for the parents, and definitely builds a better relationship both immediately and in the long term.
I think the question of whether or not to have kids should not be addressed from a financial standpoint. Instead you should ask yourself: "Is being successful in my career more important to me than raising happy children who can contribute to society?" If the answer is yes for both husband and wife (and I think a long-term, legally-committed relationship between the parents is a major factor in raising happy kids), then maybe you shouldn't have kids. No problem. The world needs dedicated people who pour all into their work. If one of the parents is willing to put raising kids above career (I don't mean to not work, just to put the kid's well being above career success), then having kids might be a great idea. In our case, initially I worked a lot and my wife spent a lot of time with the kids, now she is working a lot, and I'm spending a lot of time.
Personally, I think my wife and I made the right decision to have kids. The joy far outweighs the hassles, and more than pays back the expenses. Plus there is the potential added bonus of grandchildren!