Here is an article which details the reasons why the entire Detroit Mercy women's basketball team left, including Corcoran.
https://www.yahoo.com/sports/detroit-mercy-womens-basketball-player-100012049.html
The university has a law enforcement entity that can do background checks and people that can call around to get other info. I get that the executive recruiting companies do more than just search for candidates but there's nothing that they do that a salaried employee can't do (especially at a time of financial crisis as alleged by Miller himself). I suspect that the hiring of such a firm is more tied to the higher ups engaging in a CYA move than it does with anything else as it gives them somewhere else to point a finger if the hire blows up.