I might argue a bit with 9and7 prep school list (ie there are some who might put St Pauls, Hotchkiss, Milton above Deerfield and Lawrenceville academically ) but St Pauls does not take PGs I believe. I also have a lot of experience with the top NE prep schools and happy to explore more about individual schools by PM if you desire. But here is the main piece of advice for those who may be interested in this:
1. just like college coaches, these top schools do indeed recruit athletes. For hockey for example, the coach may get 2 PGs a year. If not recruited and without "admission support" these top prep schools are extremely competitive and one would need much higher grades and scores than your son currently possesses. Admission is VERY selective. Even super stars with top scores and grades routinely get rejected -- just like the college scene nowadays.
2. These PG student athletes have an expectation to play at least one other sport. School is rigorous. Most male PG students are athletes who were -- like your son -- injured and were not recruited with their original class year (or recruited by schools they desire), or young for their age, and need time and to be away to be mature enough etc to handle school. It sounds like you are contemplating this for all the right reasons, as there have been other posters over the years who consider a PG year because they hope it will dramatically change their recruiting process (despite not suffering an injury or setback).
3. On that last note, the timing for baseball recruiting is tricky then. Obviously, if he plays for a prep school that will not be until spring 2021 when he will have already needed to apply to college. If he has to play another sport that would be fall or winter, and in any case playing baseball in the fall is difficult, especially given the academic environments and expectations of being a resident (unless he would be a day student and you live near enough to be able to have him commute). Many schools have expectations about weekends, some have saturday classes in fact. This means that basically his big chance for recruiting would be summer 2020. We do not know his baseball measurables and where he is now after TJS but you have that information to make an informed decision about whether the recruiting would actually be any different than now by then. He may improve, but many of the spots at some of the colleges he may be eyeing would be taken. Of course there are schools who will continue to recruit a 2021 his rising senior year (notably high academic D3s, where many of the prep schools you mention send numerous players), but some D1 schools may be basically done by then, or nearly done.