The story, had you read it, points out that Angel Flight did not offer to help.
Quote: "including Angel Flight, but they weren’t able to find anybody who would fly the little girl in her serious medical condition from Houston to Arkansas."
It's more likely that Angel Flight didn't have the highly trained staff that were required to fly this young girl home successfully. Angel Flight is a volunteer organization of mostly pilots who give their time freely to fly patients needing surgery, chemotherapy, dialysis, and other treatments.
This poor young girl needed a crew of highly trained people to take care of her during the flight. Since Angel Flight is mostly volunteer pilots I don't think it's fair for your to knock Angel Flight in this case. I'm guessing their inability to help this young girl probably broke their hearts too.
I think it was probably due to the difficulty of transporting someone who is on a ventilator. I had a friend who used to be a pilot for Angel Flight until he was tragically killed in a plane crash. He always used his own private plane. Most private planes would not be equipped for transporting this type of patient.