"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.'
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.