Sick societies are like sick bodies - SFnal stories?

Are you thinking of the Foundation series by Isaac Asimov? That one is a classic. While he doesn't quite take the tack that societies are like bodies that display symptoms of health or sickness, he used the rise and decline of the Roman empire, then put it on a galactic scale, then added in a specialized branch of statistical math called "Psycho-history" that could numerically predict larger movements in that society, such as its collapse.

The rest of the series is based on his legacy, the creation of an organization called The Foundation that is designed to stand apart from the rest of the collapse of a galactic scale empire, and be a deliberate attempt like monks during the dark ages to protect and preserve the height of knowledge and science as the rest of the empire collapses in barbarism, recovers and starts trying to reconquer each other all over again.

It's such a long-established element of political rhetoric that it could have come from anywhere.

I actually suspect this begins outside of SF, with the Fisher King story from Arthurian legend. It's a little different from what you're discussing, but in the days when the Grail Cycles were being written, a king had absolute power and was considered to be the God-chosen representative of his country. The Fisher King is wounded accidentally by a knight, and can no longer do anything but fish in the river by his castle. As he suffers, not only his people but his literal land also suffers, with crops and animals dying. Only the Holy Grail is capable of healing him.

This is interesting. Were the ones who wrote the Grail Cycles the ones most invested in the idea of a hereditary King having absolute power?

Back to topic, many ancient civilizations had tales of a Golden Age and then an era of decline that happens when the people become too arrogant/secular/lazy/whatever.