What you're describing is precisely what my son did the summer prior to his Freshman year; and, since then, it's become an increasingly popular choice. I want to say that something like 6 to 8 of this year's recruited players went to summer school at his university.
It gives them the sort of head start you mentioned in the classroom, and they have the opportunity to begin using the university's facilities to train. (I'm pretty certain that none of the coaches can be involved; but, support personnel like the strength coordinator can.)
Another benefit is that it prevents them from wearing themselves out playing summer baseball before beginning Fall baseball. They need to be rested and in as good a shape as they can going into Fall baseball.
Evidently, it has no relationship to the draft. This year, a high profile recruit for both football and baseball at UNC attended summer school and began working out with the football team; only to make a quick exit on August 16th to begin his professional baseball career (...$2,750,000 richer!)
Athletic grants also don't come into play; as they're paid at the beginning of the Fall semester.
Best of luck to your son!