One usually goes to the Academy at about twelve. It’s assumed that one has been tutored at home up until that point in Shadowhunter basics.
At twelve, Will was the raggedy desperate boy who’d showed up at the door of the Institute and said he had nowhere else to go. Charlotte wasn’t going to pack him off to the Academy. Especially since he lacked the basics of knowing anything about Shadowhunters. He would have been chewed up and tortured. Charlotte didn’t know the specifics of Will’s problem, but she certainly knew enough to know he would have an awful time.
At twelve, Jem was poisoned and dying. The Academy simply would have not have wasted the resources training someone who wouldn’t survive to use the training.
The Clave are not nice people. Their school is not necessarily a nice place. :)