Private schools that did well with in-person learning last year could take measures public schools couldn't, and had much better compliance rates for things like masking and distancing. For instance, parents wouldn't let public schools suspend sports programs. There'd have been actual, physical violence if that'd been seriously attempted, lots of places. Private schools? Might be a thing they could do (and many did). Many public schools have to deal with kids who couldn't be trusted in outdoor classrooms, or face parent agitation when they take kids outside in anything but perfect weather (this is a weird phenomenon I've seen repeated at a couple private schools vs. public schools in the same area—the private-school goers want their kids out unless the weather's so bad they'll lose fingers to it even if they're bundled up, or there's an intense thunderstorm or something, while the public schools have been cowed into only taking kids out if the weather's quite good; I don't know what causes this).

Private schools could plan better because, despite supposedly being more beholden to their paying customers then public schools are to their constituent families, in practice they seem to have an easier time saying "nope, this is how we're doing it, deal with it", for whatever reason. They didn't have to pay any mind to morons trying to strip all protective measures every time "the numbers" started to look a little better, or lose planning time & focus to misguided planning for those sorts of things.

Private schools may already have had things like longer and more recesses than nearby public schools, so already spent a larger portion of the day outside—public schools have cut recess down to almost nothing, chasing those sweet, sweet test scores.

Private schools also may have had both the money and the leadership to get things like ventilation improvements or other expensive measures done fast. Public schools were crippled by indecision, and various factions pushing for things to go "back to normal" ASAP, fighting any spending or policies that acknowledged this would be a problem for more than a very small number of months and trying to deal with it in those terms.