>>24500152 (OP)"Muh rape and pillage" is a meme, but the Mongol Empire did not need to be le pillaging rape horde in order to still be absolutely non-conductive to a literary tradition. From what we know, the Empire and it's successors did establish pretty advanced and developed courts, but they had an insane military and administrative leaning, while also experiencing substantial educated manpower shortage in those courts.
There simply wasn't enough legit Mongolian Mongolians around to take even the new military and bureaucratic positions, much less create a surplus of aristocrats for some third and fourth sons to seek immortality through letters. The cutthroat, mobile and unstable nature of the courts at the time did not help - and unlike the rate and pillage, those factors are hard to exaggerate. If you were a young Mongolian aristocrat at the time of the Empire's peak, you were really busy trying to kill your older brother and not getting killed by your younger brother, instead of reading Avicenna and shit. What languages you knew you'd pursue for functionality and quantity, as through a decade you are liable to be assigned to a dozen different localities and peoples by Khan, so that you never build up a personal power base to challenge his rule, and consequentially you never develop a deeper understanding of any of those cultures.
The courts might have developed into stable and decadent enough entities to permit a flourishing written tradition in a few centuries, at least in the successor states that did not collapse like the Golden Horde, but the same shortage of officials (and simple pragmatic factors) lead to wide inclusion of local authorities, many of which already had their own written traditions, so what Mongol shit could develop instead was just subsumed by local, preexisting cultures.