I'm not saying that the US banking system is or isnt ridiculous... but in the US, anyone with a bank account can indeed send money to any other account in the world, given the account number, usually for a fee, like many other places; these are referred to often as wires, though there are actually many types of transfers. Getting it free usually requires a balance over a certain amount (banks live on fees, sadly).

Whether one can do it online depends on the bank, but most US banks and credit unions now have the ability to allow transfers and payments to anyone given a postal address (to be fair, if its just to a person, the bank may issue a paper check instead of a true electronic transfer. Sigh). They mostly call this "online bill pay" or other names, but it's become pretty full featured. In addition, most US banks are allying themselves with various person-to-person payment systems (Popmoney, though archaic and painful in use, is used by many US banks including Ally, Bank of America, TD Bank, Citibank, Fifth Third Bank, PNC, Regions, Suntrust, US Bank, and Wells Fargo, for example; Chase has it's own system as do some others. ClearXchange, Dwolla, Venmo, Paypal, and a slew of others are also gunning for this space).

So, yes, lots of problems, but I don't know if above is entirely a fair statement.