The difference isn't that Mesopotamia generated more records than other cultures. It's that Mesopotamian records become practically indestructible when burned.
(This didn't help much in Babylon itself, which didn't burn.)
WRT Babylon, I don’t know how many tablets are left undeciphered. This article (or a comment) mentions the famous cosecant tablet. We have the tools to fill in the gaps for their language and math. I think extrapolating past that is at least as ahistorical as speculating about what was lost at Alexandria.
The ideas are conceptualization, generalization, and rigor.
The Pythagorean triples are a conceptualization that for certain right triangles, the squares of the three sides obey a relationship.
The Pythagorean theorem is both 1. a generalization that that would be the case for all right triangles and 2. rigorously proved it. Furthermore, at least for the proof in The Elements, it was axiomatic rigor, proving this via a chain of logic starting from base concepts.
Now to the cultural pride aspect. Many cultures came up with concepts in mathematics. Many cultures generalized. Many cultures were rigorous.
The argument is/should really be that the Pythagorean theorem is on the far end of the rigorous, generalization axis. The better argument actually is that The Elements is.
My personal take is that the Pythagorean theorem isn't, at least in the long span of mathematics history, an exemplar of conceptualization.
And yes, there is some overlap and ambiguity with these ideas and you can argue about where the boundaries lie but I don't think we gain that much from that.
If people don't think that other cultures conceptualized, generalized, and were rigorous, just look at The Art of War, Bhagavad Gita, etc...
Babylonians had no problem arriving at solutions to square roots, and a lot of their texts are yet undeciphered. If we find that Babylonians ruminated about irrational numbers, too, then we’re talking about a major revision in the history of math. Until then, to me, the concepts in the Greek treatments are more innovative than the practical usage in Babylon.
I think your take is supported if it turns out people have read way too much into the religion around Pythagoras. I think Pythagoras would be surprised to be famous for a proof when he’d rather be famous for musical scale or some weird ritualistic dance or something. In fact, as mathematicians we might be better served by later proofs, which is implied by your other points, I think.
Recent movements to revise the recognition of Pythagoras are doing the right thing when they acknowledge the path dependence of unbroken written communication of ideas is often mistaken for awarding prestige to a particular institution, educational tradition, or culture.
If we discover a proof earlier than Thales, regardless of whether it’s Greek, we might revise the discovery of trig. But this is about certain triples being used practically in Babylon (who recorded everything) without any recorded interest in the relationship between triples.
Because Greeks first recorded the important part —- the proof —- I think it’s factual and not at all offensive to refer to Ancient Greek culture as influential in that recorded innovation in geometry.
I’ve seen a movement to broadly give credit to other cultures for Pythagoras, but knowing about triples is just not enough to say that they promoted innovation leading to trig.