The main problem with this argument and others statements similar to it is that we don't have objective evidence for the state of the world in 1200.
The information from before cheap printing is heavily biased, mostly hearsay from few individual sources that had their own agenda.
The argument you're making is sufficient to discount and any all records of history. And indeed not even history of the long past, but also the present. Do you know exactly what happened in Syria from falsifiable objective accounts, or are you relying on information spread by a small number of sources each with their own agenda?
I think your argument definitely deserves consideration when you get into the minutia of history, but when you're talking about the big scale (this empire ruled this land at this time) I think your argument is more about a disbelief in everything, historical nihilism, rather than justifiable incredulity.
"The Mongols created a single unified empire than stretched from western europe to eastern asia [...] one that was also extremely tolerant of foreign cultures"
Was there a gigantic Mongol empire? Sure. Was it unified and extremely tolerant of foreign cultures in the sense that we use these words today? Not bloody likely. What kind of tolerance leads to what you also note:
"(1 in 200 men, around the world, are direct descendants of Genghis Khan)"
Outside of religion they also even allowed local leaders to remain and organize their subjects as they saw fit (subject to things such as guarantees of religious freedom) including even maintaining their own militaries. The only requirement was a 10% tax to the Mongol empire.
They way he spread his genes was pretty much as the way we do today. The only difference is that he had hundreds of wives. Take somebody with several times the wealth of Jeff Bezos today and make him the absolute ruler of the USA, EU, and China. And now he imagine he asks, genuinely asking - not an offer one cannot refuse, practically any woman on this Earth to be one of his wives. Very few would say 'nah, I'm holding out for somebody better.'
Before we got to see what the internet actually became, it was a realm of endless idealism. It was going to be something that would finally unite people across cultures and any sort of boundaries. People would be able to share ideas, communicate instantly, and effectively turn humanity from a disparate group of people separated by arbitrary lines into simple a group of diverse and interesting people all sharing and communicating among one another. Instead we used its filtering ability to break down into even smaller groups than before. Before the internet Americans were mostly unified by an American identity. Now instead we've splintered off into a whole host of little subculture identities driven by the great ability of the internet to unite niches and expel the nonbelievers.
But this isn't about the internet. It's about people. Birmingham is one of the most "integrated" cities in England. This [1] page has heat maps showing the various religions there. There's basically no overlap. Instead of integration you have the Muslim section, the Christian section, the Jew section, etc. Even the Sikhs have there own little section, though granted they do share it a bit with the Hindus. Now instead of having one mostly unified city you now have a half dozen cities that have some very sharp cultural conflicts. The irony that literally bringing people closer together can, figuratively, do the exact opposite.
Even the notion that we're becoming more integrated is something I'd question. This [2] is a video showing the borders of the world over time. The surprising thing you'll notice is that, over time, divisions between society have actually been accelerating. Check out the mid 1200s. The Mongols created a single unified empire than stretched from western europe to eastern asia. Very near the entire known world at the time. No need for borders when there's one world government, and one that was also extremely tolerant of foreign cultures. Can you imagine living in this time and then somebody saying that within a couple of centuries the entire empire would be scattered like ashes in the wind?
And like nearly all the empire collapses you can see in that video, it was caused by internal culture clashes. Should the Mongol empire adopt western traditions and become a sedentary settled empire, or should it remain a putative Mongol empire? One side said x, the other said y, lots of people died, the empire broke up. A story told a million times on various scales. Of course even if they had chosen one direction or another, it likely would have had the same ending because once one side or another reached a critical mass - you'd see the emergence of internal insurrections.
Of course we've had a pause in open war since 1945 but nuclear deterrence isn't going to last forever. Imagine there were no nuclear weapons today. Do you think China, the Mideast, the United States, China, Russia, Europe, etc would still be maintaining the already tenuous peace we have? Without nuclear weapons the Cold War would not have been the Cold War, it would have been World War 3. And as soon as nuclear deterrence can be nullified by technological means, I expect we'll be quite quick to pick right back up.
[1] - https://en.wikipedia.org/wiki/Religion_in_Birmingham
[2] - https://www.youtube.com/watch?v=-6Wu0Q7x5D0