I only have whatever portion of your objection that you expressed to go on. You said:

A lot of the stuff they are credited for in these excerpts were really generic Asian civilization, which had its own ebb and flow and happened to reach a high point (in some ways) during the brief window of Mongol supremacy, or were particular innovations of the Song Dynasty that were exported by the Mongols.

None of which is contradicted by the connotations of the phrase "Mongol Awakening", so none of which are a reason to find the phrase grating.

I've been accused of being a Marxist, yet while Harpo's my favourite, it's Groucho I'm always quoting. Odd, that.