How can you balance your desire for the best possible academic education for your children with ensuring that they are well-rounded?

I am soon to be moving to a new city with my children. Unfortunately, the public school system in this city is not great. Meanwhile, there is a small independent school district within the city that has fantastic education. However, in order to attend the school, the family needs to live within the district, and of course house prices reflect this. As a consequence, this school district functions much like a private school because only affluent people can afford to live there. Likewise, there are several fantastic private schools that provide a great education but they suffer from the same issue as above.

My gut instinct is to provide the best education possible, yet I am concerned about the insular experience this will provide. Do I have an obligation to them to send them to the BEST schools, or will this be a long term mistake? What have others chosen when faced with this decision? Is there any hard education data on outcomes of students in similar circumstances? Can private schools produce well-rounded adults?

Please no judgmental answers to this question. Thanks.