I think it's possible for the Mongols to come close.
When they fought, especially if they fought to end a rebellion to their rule, they had a tendency to destroy irrigation systems and fill in canals. They did this mostly to allow for easier movement in case they had to return--canals and ditches were difficult for their horses to move over. But also, they didn't really trust agriculture--the wanted fields to lie fallow, to return to pasture land.
Secondly, they made a point of asking cities to surrender to them. Cities that did were largely spared (although they would have large numbers of artisans taken away, as well as treasure, etc). But cities that did not, or in times where they wanted to make a point, they were quite willing to kill everyone in the city.
So imagine a TL where the Jin and Song try to resist as long as they can, necessitating a many year long campaign on the part of the Mongols. You'd see city and city razed to the ground, with every inhabitant killed. You'd see a massive die-off due to famine and disease, too, as farmlands were systematically destroyed and farmers fled to the cities for protection. Even after the warfare had ended, it would take generations to build enough dams, irrigation channels, and such to return the land to a semblance of its previous productivity. After the Mongols marched through what is today Iran, it took 300 years for the population to recover to what it had been before. If that happened in southern (Song) China, the bread(rice?) basket of the region... the fall in population would be enormous.
When Kublai Khan established the Yuan Dynasty in 1271, the Mongols had been ruling over China for 50 years, just as Mongol territory. There were no imperial examinations during that time, or any system of Chinese rule. The Yuan Dynasty was established more or less to help the Great Khan rule over his large and restive Chinese populace. If the Chinese population was much, much smaller, I doubt he'd have bothered.
The Mongol rule would eventually end, of course. Their population was simply too small to rule over all of China indefinitely. But if, instead of a few decades of foreign-style rule, there was an interruption of a few centuries... The effect of Mongol culture on Chinese culture would be much, much stronger. The Chinese sense of history would still be strong, but you wouldn't have anyone alive who could still remember what it was like when Chinese ruled China. You might even see Chinese rulers modeling themselves on Mongol styles, the way Tamerlane did.
You can make this even worse if the Jin and Song lands are divided between different Khans after Chingis Khan's death. After the Khans begin to separate into different empires, could have the Jin and Song cultural division last a lot longer, instead of being "unified" into one Mongol Empire.
The effect of all this would be enormous. China would be weaker years down the road, too, would might lead to them being conquered outright by Manchus, Uyghurs, or even eventually European colonies.
It's not impossible to see a modern-day "China" that is a few different nations, each of them poorer than today, with a half-Mongol language and culture.