People recognized them long before the 1770s. Natural Law goes back well before the Founders, one of the main guys who wrote about it was St. Thomas Aquinas; though understanding that there is a Natural Law required further thought and reason on part of humans before understanding that there was also Natural Rights. Still, that doesn't mean the rights don't exist at all.
Others who recognized that we have Natural Rights as a gift from God include Samuel Pufendorf, Richard Cumberland, Jean J. Burlamaqui, of course John Locke and Montesquieu and many others prior and during that time. Oh yeah, Emer de Vattel also.
There are a lot of people, today, right now who don't even realize that Christianity is Reasonable. Did you know that Christianity is Reasonable?? There are a lot of things that people just simply do not know. There is a gap in their information.
"These "inalienable rights" are utterly alien to someone in China or Nigeria to this day"
Ignorantia juris non excusat.
Ignorance of the law is no excuse. Just because you were not told, just because the particular law you just broke is alien to you and now you're going to jail for breaking that law, does not mean the law does not exist.
It's just that your process for discovery took a particular path, that's all.
"why did they have to wait until the 1770s for anyone to recognize them"
(answer #2) Nobody "waited". The kings were liars, that information was withheld. You think the kings of the middle ages wanted their subjects to know that they were actually free??? F no. No king would ever want such a thing because then POOF their goes their Divine Right of stinking dictatorship. But just because kings are tyrants and nothing but gold plated tin pot dictators does not mean we should blame God for that.
"assuming that Anglo-American institutions and values represent some sort of human universal"
I make no assumptions. They can reject their rights and covet their dictators all they choose. They still have the rights though. Every person standing in North Korea, right now, has the rights. Their government usurps them. These are two very different things. The fact that the rights are usurped is proof that the rights do in fact exist.
"Certainly the Ancient Greeks and Romans, clever as they were, had no concept of these "universal rights", nor did they exist in the theocratic Holy Land in Biblical times."
The idea that lack of discovery = proof of nonexistence is a prime fallacy. Again, ignorance of the law is no excuse.
https://www.legalfix.com/articles/ignorance-of-the-law-is-no-excuse
All you're doing is trying to claim their ignorance as your own to glean as proof. That's not an excuse you can use.
The values system of checks and balances on power that produced the US Constitution had a long gestation period, starting perhaps with the Magna Carta. Again, a particular of Anglo (and later Anglo-American) culture. To say that it also applies to Australia Aborigenes or Ojibwe Indians is as absurd as claiming that the temperament system used in Western tonal music represents some sort of musical universal (i.e. it's alien to Asians used to the pentatonic scale, just like a constitutional republic is alien to most cultures).
Your thinking is analogous to somebody who believes that the internal combustion engine is something that existed a priori nature, like gravity or electromagnetism, as opposed to being the outcome of people's creative impulses.