EUR payments pass through ECB. It's possible that a small Taiwanese or Japanese bank might not have correspondent accounts with a European bank directly. If they for example transact heavily in USD and very little in EUR, they might find it easier to simply use their US correspondent bank for EUR as well. In that case, the EUR transaction physically happens in Europe but US regulations are involved by way of the US correspondent bank.
But that's not the norm. Major Taiwanese or Japanese banks have their own branches or subsidiaries in EU for EUR clearing and/or have correspondent accounts with EU banks, in which case US would not be involved in the transaction at all.