Dick Cheney Hunting Accident

As described in "Secret Service describes medical response to Cheney shooting", Cheney's medical crew made the decision to take shooting victim Harry Whittington to Christus Spohn Hospital Kleberg in Kingsville. Note that that's a smaller hospital than Christus Spohn Hospital Memorial, which is the area's trauma center.

So, why was Whittington taken to the smaller hospital rather than the trauma center? Could it be because it was a bit less "visible"? Would it be easier to hide the victim's condition?

An apparent doctor raises questions here. Are they trying to hide something? And, if so, what?