Because, damn, did Trope Namer Syndrome strike with a vengeance here. We're lucky Grey's Anatomy is still on the air and thus still doing backstrokes in the Small Reference Pool.
Okay, cleverness aside, this needs a more generic name because Grey's Anatomy certainly did not invent this trope (as anyone who watched Emergency in the 1970s can tell you). Maybe we can just trim off the show name and call it "Emergency Medical Response"?