Should I switch my son from public to private school because of the teacher shortage?

That really depends on your financial situation and how your son's learning is actually going at the public school right now, because switching just because of a teacher shortage doesn't guarantee that a private school will be better if you don't first check what's actually happening in his specific classroom. Before spending more money, it's probably worth talking to the administrators about how they're covering those absences and seeing if there are improvements on the horizon, or at least if the school is being honest about the problem.