It all depends on the private schools. Public and private schools vary a lot.

My kids go to private school. We live in California, and the schools here are generally pretty lousy. For example:

1. The day ends roughly one hour earlier than the private schools, except on Wednesdays when it ends roughly two hours earlier. Plus they have minimum days most months, when school closes at lunchtime. And they've had furlough days this year when school was closed because of budget problems. I once calculated that all this stuff amounted to 4-5 weeks of lost instructional time compared to the private schools. This is huge.

Is this the case where you live?

2. The private schools around here use all that extra time for music lessons, art lessons, field trips, and foreign language.

3. Around here, the private schools are much more open to the idea of ability grouping and grade skips. I have three skips between two kids that never would have happened in public schools (I called around and asked and got some very anti-skip responses from local principals).

Private schools can be averse to skips too, but you have leverage because you don't have to send your kids there. You can always negotiate at an interview ("We're thinking of sending our child to this school; we think he would benefit from a grade skip. What's your policy?" Then sit back and let them talk.).

4. There are obviously bad private schools (e.g. The earth was created 6,000 years ago!!) and good public schools (e.g. Davidson Academy for this crowd).

I may come across as being very anti-public schools, but I'm not. I went to a public school in New Hampshire that offered everything I listed in point 2 and then some. They ability grouped starting in grade 1 and were open to skips. That school was great. AFAIK, it still has a lot of that stuff.

I don't like spending a small fortune on private school fees, but around here, we simply have no choice. It's not that the private schools are a perfect fit, either, but for us, they're a much better choice.