This is an elite boarding school in the Berkshires. Not sure that the cell phone ban is the most important thing in these outcomes and I'd hesitate to draw sweeping generalizations here.
Private schools, perhaps counterintuitively, have greater ability to enforce rules that some parents dislike, than public schools. They can organize around and enforce a general culture in ways that public schools cannot, and this ability increases the more prestigious they are. Families self-select to be generally agreeable to and aligned with the school's culture, and bad fits can simply be sent away. Public schools have to deal with whatever mix of families they happen to serve, period. Private schools are sometimes pointed to as being more-accountable to parents, and that's sort of true, but they also have much greater freedom to tell a loud minority of their "customers" to, frankly, go eff themselves.
An observation of trends I've seen with a limited dataset but that I suspect generalizes pretty well:
Better-off, if you will, parents, or at least those with a "higher-class" cultural background (Fussell's upper-middle and higher—doctors, lawyers, executives, in short) are giving their kids phones later, restricting use more, and are more-amenable to restrictions on phones imposed by schools, than lower-class (Fussell's middle and lower) parents. These factors tend to significantly reduce student phone use in ritzy private schools, "captive" nigh-private-school public schools, and the tip-top of other upper-tier public schools, compared with practically all the rest of the school landscape.
I fully expect this to widen already-large class differences in education.