There is a relation between silver shortage and silver coinage, but it is a one way street: when there's no silver, you obviously can't make silver coins. However, when no silver coins are minted, it doesn't automatically follow that there's no silver. That relationship is disturbed by such things as demand for silver, regulation, war and other political circumstances and even technological knowledge, but most of all regulation.
Moreover, a shortage of silver actually means a relative change in the situation, usually (but not always) expansion of trade and economic growth. The silver is still there, but it is not enough to accommodate the needs of trade. This was the case in Newton's time, causing a great recoinage. It can also mean a structural deficit in the balance of payments. That was the case in the Kipper und Wipperzeit in Germany. Silver had flowed to countries with a trade surplus with the empire.
The situation in the UK in 1776 was different. The silver was there, but the mint had lost its traditional access channel: The surrender of bullion under their licenses by te native mining companies ceased about 1750. When a smart solicitor discovered that their obligatory offer of silver to the Mint need not be made below the market price, this source entirely dried up
*. In the next half century there was almost no silver coinage, apart from 1757-1758 and 1787, when the Bank of England ordered coins to be used as Christmas gifts for their customers. Furthermore, a small amount of coins dated 1762-1763 was produced from "a treasure taken at sea" including shillings for Lord Northumberland. The rest was maundy money. In short: the shortage of silver coin was due to regulation that forced the Mint to buy silver below market price. Unsurprisingly, it found few sellers.
So was there a shortage of silver in the 13th century? Do some superficial Google searching and you will find that the answer is yes. In England. I found no information on Central Asia. The English situation cannot be extrapolated to Central Asia, if only because the English economy was highly unimportant at that time, while Central Asia was still a major benefactor of the silk road trade. If there was any influence, it went from Central Asia to England, but even that is unlikely.
EWC has mused in earlier posts that fiduciary coinage led to a silver surplus, which drove silver into Europe, in exchange for (Slavic?) slaves. Maybe. Not impossible. That scenario implies implicitly that the price of silver (in terms of slaves) had fallen and that the fiduciary coins were accepted as tokens for their value in silver. Not impossible, but it implies in turn that the rulers of Central Asia retained enough silver to pay foreigners insisting on silver and that they never fell for the temptation to produce too much "free" money, which would have caused a disastrous loss of confidence in the fiduciary coins. Since the Qarakanid coppers seem relatively plentyful, chances are that the experiment failed, just like that of Mohammed Tughluq's forced currency
in India. Both experiments may well have been patterned on Chinese cash coins, which obviated the use of silver coin in China (though it did use silver bullion to some extent.)
* Sir John Craig, The Mint
, ISBN 9780521170772, page 246.