Why must non-US transactions often clear through US banks?

If the clearing and settlement is done in United States Dollars, even for a transaction that is not originating or terminating in the United States, but in countries outside the US, then the US banking would still be applicable.

For all transactions related to the US Dollars, respondent banks are responsible for clearing these transactions through authorized correspondent banks in the United States.

Think of any transaction done on the US Dollar as a cheque that must be cashed by a US bank and this is why even if the cheque is being handed from Country A to Country B, the United States banking needs to be involved to clear it, as the ultimate ledgers of the US Dollars are being maintained in the US and not abroad.