Hahahahah you're bubbling with inspiration, aren't you. :3 

Since the Academy's history is more fleshed out, and keeps being fleshed out all the time anyway, we've established that in it's current state it was established 178 years ago more like a bootcamp and a training facility than an Art school (before that it was a mundane arts school for some 50-100 years before it was ruined shortly before Judas' rise to power).

Also, back then the instruments were also randomly bonding to people and not given by Hazumi and the Academy's role was to find and train the ones who'd bonded. 

Thus. If you want, you're welcome to write flashbacks about it (your scenario seems very likely) and even play a guest-character in the Mending season (the prequel running parallel to this one but taking place 178 years prior.)