[shamir]

Actually, there is a system in place for any bank to transfer money to another account at another bank. Its called ACH (http://en.wikipedia.org/wiki/Automated_Clearing_House), and thats how direct debit and most other interbank transfers in the US work. Internationally there is SWIFT and wire transfers, but within the US all that you really need is a name, account number, and bank routing number. All of this information is printed on a personalized check from any bank; thats why some people ask for a void check to setup auto-pay.

That said banks in the US have been very very slow to make this technology easy for customer's to use, partially because they just suck at user friendly technology, and partly because they don't want to make it too easy for people to move their money to an account at another bank.