Sorry to butt in here, but I must disagree with this statement (I assume said lecturer wasn't a historian of the Yuan dynasty. Neither am I, I worked on Mongol rule in Central Asia and the Middle East, but sources based on Yuan documents were essential to us). The Great Khanate in China (I.E. The Yuan), was indeed founded after a series of bloody Mongol victories, and many Chinese mandarins did meat their deaths, but those that survived - by all real estimates (those not based on either Ming or the Mongols own, self-aggrandising propaganda), - the majority of them, were quickly taken on by the Mongol and Turkic lords, and quickly were back up to scratch, taxing, recording, and administering as they did before. The Mongols even attempted to create a written version of their language, but it didn't catch on. Instead of not writing and reading however, they just began learning Chinese. Several Mongol leaders, in Persia, Mongolia and China, were able to speak the local languages to some extent, though they often used Mongolian in public, to keep their thoughts relatively secret and to draw a line under the fact that they were in charge.
So why so few documents? This is a much more difficult question. Partially, the Yuan dynasty was relatively short lived (less than a century, less than that if you only count the most stable period), and thus simply didn't have a chance to produce as much as their successors, the Ming, who lasted about 300z
Secondly, the Ming, much like the Persians, did not like to admit that a foreign power had ruled for such a long period (and in some ways had actually unified large parts of the empire for the first time). They seem to have destroyed records and claimed achievements for themselves (my favorite example is the 'Ming era' observatory that still stands in Beijing, it is an almost carbon copy of one that was built in Marageh Iran under the auspices of Hulegu Khan and Nasir Al-din Tusi. The Ming (or, just as possibly, the current Chinese cultural administration) reclaimed it as their own in their guidebooks and museum.
OP's central thesis is possible, we don't know exactly why we we have so few administrative documents, but we're quite sure they existed (plenty of etched bronze paiza passports exist, and tons of remarkably detailed agricultural and tax records from neighboring khanates). I should also mention there may be many more records from the Chinese Mongols than I know about, due to my focus. Because pretty much no university (I don't think any) offers BAs or MAs specifically in this stuff, many otherwise experts on China, the Middle East or Russia know practically nothing about this century/these centuries.
See David Morgan's The Mongols for an overview, and for the observatory in Beijing, either book by George Lane, or a book whose name I can't recall (sorry, out on a bus!) by George Saliba.