The Fed has had a wire service (Fedwire) for banks, allowing them to transfer their balances on the Fed’s balance sheet to another bank during settlement, since before the dollar moved off the gold standard. It was initially done with literal telegraphs - not sure at what point it became digital.

It obviously has no pseudo anonymity, is literally the least democratized banking system in existence, and is subject to the government’s whims in a whole host of ways. But it is a digital ledger of massive sums of real dollars (the banks can ask for it in cash if need be), and you couldn’t really steal the money even if you managed to create an unauthorized transfer on some bank’s master account.

Most states have in fact invented bank transfers for that purpose.

The states (or rather the national banks of said states) are usually the ones running the central clearing system. That's the place where all the different banks report their net change in relation to all the other banks, and settle that change on their account with the central bank.

Believe it or not, banks don't ferry around cash to each other. It's all just numbers in a computer.