Iatrocracy…is a perfectly good word, specially coined for the purposes of this paragraph. It comes from the Greek word for doctor, iatros, and it means government by physicians.
Why not? Who is qualified to keep the world alive and in good health if not the health specialists? Let Iatrocracy have a chance to show what it can do along with autocracy and aristocracy and democracy and mobocracy and theocracy and bureaucracy and gyneocracy and biocracy and plutocracy, and so all the way down to idiosyncrasy.