> many private schools don't have that much additional capacity
In terms of teachers, maybe not, but in terms of facilities, any school that was around in the 70s-80s should have plenty of classrooms and lockers. And I imagine that as public schools shrink, there will be teachers looking at those jobs.
The public schools around here are having a serious staffing crisis because wages, which were already low and stagnant for years, didn't go up as wage inflation hit the rest of the economy, and teachers are sick of the whole thing and burnt out, so are leaving. There's a big "leave for literally anywhere else—you'll be better-paid, less stressed, and more-respected" movement among teachers right now, because all that's more-or-less true.
In terms of teachers, maybe not, but in terms of facilities, any school that was around in the 70s-80s should have plenty of classrooms and lockers. And I imagine that as public schools shrink, there will be teachers looking at those jobs.