That is not how SWIFT works at all. SWIFT is basically a peer to peer financial network between banks. Any bank can send any amount of money to any other bank, without any kind of approval from the US, provided they have an appropriate common facility. Which could be anything from both banks having accounts at a different common bank, or both banks holding accounts at a common national bank, which allows them to transact directly.
Pretty sure I saw an article describing some approval process. While trying to find it I found references to FATCA, TFTP, and some reporting by the NYT