You ask a very interesting question. First I want to state, that this depends upon many factors other than just school. Does moving involve a better or a worse quality of life in all. There is more to life than school. Would moving involve a longer commute, near any places of interest, higher cost of living, family friendly neighborhood...? We just recently moved from a small city, White Plains, NY to 40 minutes north. We now have more space, more value for our home (bigger home, brand new we built it), but our commute is longer. Schools weren't an issue as we thought the White Plains Public Schools were fine. Schools are about the same maybe a bit better in our new community, but not as diverse. Diversity is something that I feel is a good thing as it is the real world we live in and one day our children will work in.
Now getting to your school question, again I think for me it would depend upon the two schools I'm comparing. Why would I leave, are the teachers terrible, the scores very poor? I don't think sending children to a religious private school is generally better. They have a high teacher turn over rate and are very dependent upon income from student tuition and their church. So, they can close when times get bad. As a teacher myself, I do believe our public schools generally work fine. They tend to offer a more variety of courses, programs, and activities. Now, I know some are real problematic. If that was the case, then I would not hesitate to move as that could mean in general my quality of life would improve. Sending them to a private school may just not be enough. If you can afford to move to a better area, then perhaps that would be best. Sometimes moving older children may be a traumatic experience so that too would have to be a factor.
Also something to consider, two of my best friends went to Catholic school all their lives. They only made one friend as their parents didn't feel like driving them all over the county for play dates. This friend happen to live in the same town.
Sorry if I am rambling on and on, I need to go get a cup of coffee