I do not know if the Mongols could have subjugated Europe, but there were several factors working against them. Tactically at the time, nobody had beaten them in open battle. The only major battle I know of between the European powers (28,000 men) and the Mongols (less than 20,000 men) resulted in a crushing defeat of the European powers. Two other points.

But the difference is that China was on the Mongolian border and had much territory which was ideal for horsed armies.
The western 2/3 of China is rugged country, much of it still today sparsely populated. Not good country to fight in at all. Still, most of the population was/is on the costal area. Perhaps it was Russia that was intended.

Europe was 4000 miles away, heavily wooded, and with little if any pastureland to support the large numbers of horses the Mongols employed.
I can only scratch my head, since Europe at that time was a very agrarian economy, with a polulation of about 60 million, and if grassland/pasture did not exist to sustain ponies used to subsisting on steppe grass, Europe could not have supported half that number of people.

As a subset of this, the Mongols were always strategically, and often tactically, outnumbered by their enemies. The Europeans of 1150-1300 could not conceive of an enemy capable of moving from the shores of the Baltic sea to the Carpathian mountains in weeks rather than months. It was believed, at the time (~1241), that the Mongol forces in Europe alone exceeded 500,000, when the population of Mongolia at the time was less than half that. The Mongol "Horde" was a creation of their enemies imagination, and one that the Mongols did not discourage.

The main problems facing the Mongols were logistics and the nature of tribal succession, which caused more problems than their external foes did for several decades.