i think the controversy comes from you saying its common for steppe people to be either side of steppe at same time, when in your example its only the mongols for the same people.Well the most obvious example is the mongols, They invaded keivan rus and china around the same time.
Another example actually is the Huns, not the huns themselves but the Rouran Kaghnate which pushed the huns west. According to the huns as they entered europe the Rouran Khaganate was hot on their heels, meanwhile there are Chinese sources thousands of miles away contemporaneously interacting with them.
The Turks likewise shattered into many different culturally related peoples from china to central asia and the near east. Whether the Turks are the same people of the Xiongnu is controversial but also speaks to this.
Depending on how much one puts stock into Herodotus and other ancient sources there is also something to be said about the scythians and the medes, the later supposedly ejecting the former out of asia before eventually finally settling in Iran.
Again what speaks to the geographic factor here is just how common a nomadic people group will be interacting with europeans and chinese on opposite sides of the worlds largest continent contemporaneously. Compare this to europe and china through india before the age of sail. I am a bit surprised how controversial this idea seems to be. It is a giant flat plain across the worlds largest continent, of course it is going to accelerate the movement of the people most suited to living there more than uncharted oceans, perilous mountains in iran, the deserts of Arabia and the dense civilization of india. The silk road was so valuable for a reason, it took time, labour and resources to maintain that one stable route from europe to china through india. Meanwhile a steppe tribe could make the trip unencumbered at the speed of a horse over flatlands.
Aditionally whilst it is a vast flat land, its still hard to project force across the steppe due to its low rainfall