Well not always necessarily a 'banker's bank'. The FRB is a banker's bank. The FRB is basically a correspondent system for banks--think correspondent as in 'pen pals'. By having accounts at a common FI (the FRB) banks can 'correspond' financially. Any bank that offers correspondent services might offer international or domestic correspondent services. Say you have an account with Tiny Bank in Arkansas. That bank might have a correspondent relationship with Bank of New York whereby someone can deposit into your account at Bank of New York as if it was done directly Tiny Bank. In other words you don't necessarily need to have an account at multiple banks to pull some things off and its possible to have an 'dummy account' at BoNY that BoNY that will cause deposits at BoNY to be deposited at Tiny Bank through the correspondent relationship. Basically one bank might provide services to yet another bank and vice versa. They simply correspond. If you look into the history of banks and credit cards, you'll likely see that money and post or courier services were usually 'married'. Look at Chase and Wells Fargo history. Banks provide courier services among other services or so-called services (if obstructing commerce is a service).
Of course Visa, Mastercard or Western Union would love for you to think that you always need them. The catch of course can be in the fees. If your transactions are frequent or high-dollar enough it might not matter. Are you surprised that it might cost a bank only $.20 cents to send a Wire Transfer but they charge $20 for the service?
Also MAJOR TIP: it is often easier to open an overseas account at a CORRESPONDENT OF YOUR BANK than any other bank. Because in a sense because that your bank has an account with that bank you already have an account. But no one told you--except me. And Captain Obvious would be pleading to interject that once you do that, it would be very very easy to fund that account through the CORRESPONDENT RELATIONSHIP.
Someone just anyone could just for fun go to their bank and ask for a list of their foreign/international correspondents and just give a general report here. There used to be a book that banks kept on hand that lists ALL KNOWN bank's correspondents in the world-thick like a phonebook. When you ask they might ask you to specify the country. You could ask for China. Ask them to give you all details: typically name of bank, IBAN, SWIFT, box service info, etc. You can even ask them for domestic (US) correspondents. That information allows you to see what banks your bank is connected to throughout the USA and throughout "the world".
Consider that this information is helpful if you travel and want to deposit a check into your USA account while you are overseas or vice versa. The information has always been there but...they weren't necessarily motivated to tell you...and perhaps you weren't all that motivated to ask.