No, that is not how banks work. You are not holding actual legal tender, your bank account is a credit with that specific bank that the bank promises to pay back to you with legal tender on demand. But the bank has lent out that money and might not be able to actually pay you back (or might need to be bailed out by the FDIC to pay you back if your account is under $250k).

I agree though that existing banks could technically implement costless transfers or accounts for people outside the US with existing tech, but it is not worth it because of regulatory compliance issues and the archaic nature of the banking systems. But if those issues were to be fixed, might as well just hold what are now checking accounts directly with the government, rather than have the convoluted system of banks + massive regulation + FDIC insurance.