Using head-hunters (3 party companies) to find/screen potential applicants is used by many universities (mine included) - but mainly for high-profile jobs. In fact, in my career, I have been contacted twice by such companies on behalf of universities looking for hire a dean of science (Cal Poly-Pomona and someplace in Florida - said no to both).
I do agree that if UTEP offered Stull an extension and were surprised he declined, then they'd be a bit behind. Usually you want to have the search committee in place before the start of the school year.
Hope it works out well.
lieb