I agree with the time compression illusion, although in fact we depend on point events - battles, etc - to place a temporal framework on this sort of thing that has its own illusion of granularity that isn't really justified by the quality of record-keeping at the time. These days we're better at that, but we still have the challenge of communicating that information forward unmarred by historical revisionism. And there's an awful lot of that about.
You are quite correct in most of what you say.
However, I would contend that the Mongols were very different from the Huns, Scythians and other previous steppe nomad conquerors. Most of them were nomadic tribes that moved more as peoples than as armies or as state actors with a strategic plan of conquest.
The Mongols, OTOH, were brilliant at military and political strategy and had a definite plan for conquering the rest of the world. They moved like lightning across the steppe and most other tribes could not get away in time, leaving them the options of annihilation or submission and joining the Mongol armies.
While there were of course various refugee groups fleeing the Mongols, I don’t think there was anything like the westward billiard ball effect lasting a century or more that was created by the Han defeat of the Hsiung-nu (sp?).
Looked up the date. The Mongol invasion of Khwarezm started in 1220. That is the reasonable date for Mongol activities in central and western Asia, as previously they had been limited to the other side of the Pamirs.
Interestingly, Jerusalem was sacked by a refugee Khwarezmian army in 1244, with the Christians massacred and the Jews driven out.