Whenever I get into this, I come back to the political genius in the 4th century who turned the Roman Empire into the Roman Catholic Church.

Rome continued to rule the world for a ~15 centuries. There is still a sizable chunk of the global population that obeys Rome, even though they are not ruled by anything like the Roman Empire.

Realising that the military empire wasn't going to work any more, and changing Rome's historically laisse-faire attitude to religion into an effective method of rule... that's genius at work.

The millenary histories of the Egyptian and Chinese empires disagree. Yes, dynasties came and went, but those polities endured for thousands of years—indeed one could argue that the ongoing governance of China by the Communist Party is just the latest dynastic swing.

Chinese dynasties got overthrown pretty regularly every two or three hundred years, with a good amount of divided kingdoms, or foreign invaders ruling over much of China, thrown in between.

Oh, also switching dynasties frequently resulted in the entire previous royal family being summarily executed.

For a brief(?) summary: https://en.wikipedia.org/wiki/Dynasties_in_Chinese_history

And cultural knowledge:

Burning of books and burying of scholars, Qin dynasty, 213 BCE

https://en.wikipedia.org/wiki/Burning_of_books_and_burying_o...

Xianyang Palace Archives

https://en.wikipedia.org/wiki/Xianyang#History

I think we’re talking about forms of government and not geographical countries. China definitely doesn’t fit here since it’s governing bodies have shifted so drastically in the last few hundred years. I don’t know much about the specifics of ancient Egyptian government but I imagine it would vary with the dynasty if not with the individual ruler. Traditional monarchies are based on the whims of a ruling elite, family, and individual after all. They have no constitution to provide continuity and all law, secular and religious, is superseded by royal decree.

I'll have to read up on Egypt and pre-XX-century China. How oppressive were these dynasties typically? What did the life look like for a regular subject?

We'll probably never know, unfortunately. The literate classes in ancient societies rarely wrote about the lives of subsistence farmers, who were the vast majority of the population, so while some details can be reconstructed it's very hard to speculate about how oppressed a typical farmer would have felt.

The Vatican has almost a couple of millennia of continuance, but I'm not sure I'd want to say continual stability per se.

Yes, for any "couple of" smaller two. And more than once it was not clear whether the Vatican was still relevant at all.

The Vatican is also not the same size and breadth as the US or China.

The Vatican as a country, I would agree. The Vatican as the Catholic Church, does seem to be on-par with those, at least in terms of head count.

Compared to today's bureaucracy, taxes, etc, they were free as birds...