I know some private schools are poor as well, but I wonder how much has to do with size. When we moved to a different state my brother was entering 10th grade. My parents deliberately moved to one of the top districts in the area (a large district with a large high school). My mom took him to register at the high school and they could not offer him accelerated courses that he needed. My mother immediately drove us to a very small Christian high school (only 40-75 kids per grade) and they were able to tailor instruction to him. The teachers worked with him one-on-one. They sent him to the university for classes as needed. While there were very few in the "gifted" category, he did great, and went on and became a physicist.
Seems like large schools/districts have so much red tape and bureaucracy and they try hard to fit everyone in a box because it's too complicated otherwise.