The answer is banks, anon. Banks used to offer this kind of service by virtue of their existence. When you write a check or use a card to make a payment, you are authorizing the bank to remit the amount to the relevant party.
A dollar bill was just a claim check against an amount of silver. Most people traded the claim checks and only on occasion would a large business show up to actually claim the physical.
This was the purpose of banks and the treasury.
As for the subsequent discussion of blockchain currencies - while I do generally think blockchain technologies have loads of potential, I do not think it wise to place our currency onto them without substantial experience in the wild with them.
I am generally a proponent of monero as a far better platform for a currency (and other blockchain related systems - I have been looking at using them as a way of creating a civil intelligence system that allows information to be contributed securely but that also can be verified for its integrity and will auto-declassify under the right conditions, for example) - but think we are generally just scratching the surface of applied concepts using it.