Ur may not have been the first city, but it was the first one we know of that wasn't part of a false dawn - one whose culture and technologies did demonstrably spread to other areas. It was the flashpoint.

A contrary view — and I'm stating this deliberately rather strongly to make the point vivid:

"False dawn" is a retrospective view; which is to say an anachronistic one; which is to say a mythical one. And myths are written by the victors.

It's true that we perceive more continuity from Ur to today's civilization than from Xyz (some other ancient "dawn of civilization" point) to today. But why? Surely in part because the Sumerians and their Akkadian and Babylonian successors were good at scattering their enemies, killing their scribes, destroying their records, and stealing credit for their innovations. Just as each new civilization claimed that their god had created the world and invented morality, each claimed that their clever forefather had invented agriculture, writing, and tactics. If the Xyzzites had won, they would have done the same.

What's the evidence? Just that that's how civilizations — particularly religious empires — have generally behaved since then. The Hebrews, Catholics, and Muslims, for instance, were all at one time or another pretty big on wiping out their rivals' history and making them out to be barbaric, demonic, subhuman assholes — when they weren't just mass-murdering them. So our prior for the behavior of the Sumerians should be that they were unremarkable in this regard; they did the same wiping-out of rivals' records that the conquistadors and the Taliban did.

Today we have anti-censorship memes; the idea that anyone who would burn books is a villain and an enemy of everyone. But we also have the idea that mass censorship and extirpation of history is "Orwellian" — as if it had been invented in the '40s! This is backwards; it's anti-censorship that is the new, weird idea. Censorship is the normal behavior of normal rulers and normal priests throughout normal history.

Damnatio memoriae, censorship, burning the libraries (of Alexandria or the Yucatán), forcible conversion & assimilation — or just mass murder — are effective ways to make the other guy's civilization into a "false dawn". Since civilizations prior to widespread literacy (and many after it) routinely destroyed the records and lore of their rivals; we should expect that the first X that we have records of is quite certainly not the first X that existed, especially if its lore makes a big deal of claiming that it is.

Put another way — quite a lot of history is really a species of creationism, misrepresenting a selective process as a creative one. So we should not look to "the first city" for unique founding properties of civilization, since it wasn't the first and didn't have any; it was just the conquering power that happened to end up on top.