Welcome to the site. It varies by school but many schools assign players to the summer program and, generally, the players go where the coach suggests. Upperclassmen usually do the same. If a player has a good option set up close to home, some coaches will be fine with that. Typically, the college coaches have a pipeline of summer programs they have relationships with where they know the player will get roughly the agreed upon opportunity/playing time and, for pitchers, innings limits will be enforced.
Regarding post-season play/Spring, most established summer teams will have players on short term contract to play the beginning of the season while the full term contract players are finishing out any post season play. There is some roster movement for a variety of other reasons - injury, innings pitched in-season, internship opportunities, performance didn't meet expectations, etc. Still, if the player is a freshman and assigned to a decent league, that is a good thing. You said the coach "would like to see him play on... " Is he assigning him or just suggesting?
Also be aware that there is plenty more information you can search on this site regarding summer college baseball.