You are correct.
But they sure enhanced the methodology.
I beg to differ, depending on how you define "enhancement."
The Nazi genocide is generally considered, for reasons I find unconvincing, a masterpiece of efficiency. My personal opinion is that it was a classic example of governmental inefficiency.
The Nazis shipped Jews and others from all over Europe to Auschwitz and other camps to be killed, using transportation facilities desperately needed for the war effort. Making the entire process wildly inefficient for that reason alone.
It is generally believed that approximately 1.1M people were murdered at Auschwitz over the course of the roughly three years of its operation.
Let's compare this to the murder methods of the Mongols. On multiple occasions they captured a city, tied up 500k prisoners, and on a given signal the 100k men in the army chopped heads.
That means that 100k Mongols managed to murder half as many people in a few minutes, certainly no more than an hour, as the 15M Germans and allies managed to kill at Auschwitz over three years.
Modern methods of genocide, as used by the Turks, Nazis, Soviets and others are often more "industrial" in nature. But IMO they were much less efficient than those used by the Mongols in the 13th century.
The Nazis murdered roughly 12M people, perhaps 5% of those under their control. By contrast, the Mongols probably murdered 10% of the population of the entire world at the time.