But, assuming the mongols would have stayed united and determined in that goal, they could have conquered Europe in piecemeal fashion quite rapidly. The Mongols would have needed to exterminate the agricultural population along a corridor that their armies will take next summer to guarantee a plentiful supply of fodder for their horses. Europe didn't have any significant military or economic advantage in this period: European advantages arose when Europeans began to master new capital-intensive innovations after 1550s.
As you admit, Europe had
people. Plenty of people. That's a huge military and economic advantage against an invader that absolutely cannot move large numbers into the territory. There simply was no way for the mongols to transfer large armies across the wastelands of Asia to attack Europe, even if they did tried to leverage on their chinese resources. They'd have to conquer
and submit large numbers of europeans, then use them to attack the rest of Europe. It was not going to happen: they never even managed to subdue the princes of the Rus, only get the occasional tribute mixed with rebellions. Europe, unlike northern China, was
too fragmented and
too foreign to be conquered and then controlled by the mongols. The mongols who did get there across Asia as reinforcements would be constantly ground down fighting rebellions.
The mongols, like the Romans, the Arabs, or the Ottomans at the maximum extent of their empires, were pushing the limits of what with their technology could be controlled by a single imperial center, or even a single people, without losing its cohesiveness,
without losing the ability of suppression rebellions. Their limits were larger because their mobility (nomads on horseback -> faster communications) was greater than the military of those other empires, but they had reached them in eastern Europe, in the territory of what became the Golden Horde.
It is proof of their military success that they did reach those limits (not that I'm accepting that the mongols would have fared well fighting in Germany, for example), but logistics were always the ultimate limitation on empires with successful armies. Even in the modern age: the planners of the british empire, in the steam age, spent much of their time obsessed about ways to cut expenses and prop up puppets to alleviate over-stretched resources (men and finances) after they bit too much by annexing the Middle East after WW1.