Well there's a couple of obvious reasons. The first are the legal waters of HIPAA and FERPA, where release of specific medical information can be limited or restricted. And the second is, if they're planning on playing Arch anyway while he's hurt, why give opposing teams any more information than necessary?
Players play hurt all the time. Last season Quinn Ewers played the entire season with a torn oblique and the coaching staff covered for it the entire time.
It was just the bizarre way he did it, that's the problem. He should have just coach-speaked it away like happens 99.9% of the time.