There's nothing about books being transfered to Kazan in either of these answers, and i've also never heard of such an action being taken. It seems a bit strange for all these books to be transferred from Baghdad to Kazan for what appers (at least to me) no discernible reason. It wouldn't have been a particularly important city in the wider Mongol Empire. It did become more important under the Golden Horde but this was several years after Baghdad was sacked.

I also have to raise issue with "an alibi being crafted to create excuses for improvement ceasing", as there was no real reduction in "improvement" (however you would quantify a reduction in improvements). I've never seen this claim by any "Middle-Eastern Historians" either (i'm not aware of any at least) . The first answer I linked below mentions Nasir-al-Din al-Tusi, who was a Persian scholar who lived through this sack, and indeed died in Baghdad several decades after its sack. His work did not cease because Baghdad happened to be sacked. Others like Al Shirazi continued their work in a Mongol controlled Iran.

It also ignores the fact that much of the Middle East was never taken by the Mongols, Cairo was an extremely important city under the control of the Mamluk Sultanate but was never taken by the Mongols. People lke Ibn Al Nafis lived and died in this city. Ibn Khaldun and Al-Shatir all appeared in the centuries after the sack of Baghdad.

This answer by u/rakony talks about the sack, also answers many questions under his initial comment that you might be interested in.

This answer by u/artorrex talks about Baghdad being in decline long before the Mongols arrived.