Not really. Genghis Khan's grandsons split the empire apart; only the part of it that was in China was focused in China. Everyone focused on the territory they controlled, for the obvious reasons.
https://en.wikipedia.org/wiki/Division_of_the_Mongol_Empire
The reason they paid no attention to Western Europe is very straightforward; the Mongols never reached Western Europe. But they reached Eastern Europe and they stayed there for centuries.
But what I wanted to highlight was that the most prized territory were in the east.
For instance in the division you linked the supposed ruler on the empire after Gengis Khan dead was a Yuan, sitting in Beijing and holding the Title or khan of khan. ( even if that did not worked out so well )
Anyway; I’m not an expert ,by far. I mostly regurgitate bits from the Dan Carlin show on the subject.
I remember him putting a emphase on how the Mongols were turned East first, culturally and in intends, and only eventually west.
( great in depth série, 10h or so long, even if I can’t stand his bombastic style )
So what? We also know they never reached either of those places. It's easy to have maps.
> For instance in the division you linked the supposed ruler on the empire after Gengis Khan dead was a Yuan, sitting in Beijing
Again, so what? A traditional title of the king of Assyria was "King of the Universe". It coexisted with the much less aspirational title "King of Assyria". But it didn't mean anything. Kublai Khan held a title that implied he held authority over other divisions of the Mongols.[1] And in fact he held no such authority, and nobody believed that he did. And...?
[1] Though that title wasn't just "Khan"... everyone in charge of anything was called Khan. What title do you think the head of the Il-khanate held?
Those were prepping for a crusade, and got surprised by the mongols arrival.
They lost pretty badly. When the mongols did setup shop and administration, it sound like their focus was China. And not the backwater of Western Europe.