You're assuming that the hypothetical university personnel responsible for the cleanup of the mess thought the whole thing through clearly and weren't in hardcore firefighting mode and reacting in ways that would seem to make sense at the time but didn't take all the facts into account. I see no reason to assume that level of intentionality.

One thing the Streisand effect illuminates is a certain type of person is not fully able to model the consequences of trying to keep information secret. It's absolutely plausible that someone in the university administration actually believed that they could tell the FBI "please keep this secret in order to protect our reputation" and that their request would outweight the agency's other duties. There exist people that are that self-important.