The summary is a bit sensationalist. Being a resident of Indianapolis, I know for a fact that there are a ton of hospitals around this area. Chances are St. Vincent's got a lot of those patients. I'm certain that Methodist would not have turned away any patients that they were not absolutely certain would receive adequate aid at another hospital, or if they thought that the patient in question was in no condition to be re-routed.
As for paper vs. electronic records, hospitals keep both. The point is that paper records take a lot longer to manage, and if they can safely do so, it's in everyone's best interest for them to send patients to other hospitals in order to get caught up on paperwork. If their paperwork keeps piling up, the chances of losing important data increase by a large margin, and that's bad for all parties involved.
No, I say that Methodist made the right call here.