So I say to my employer, well I don't have a Bitcoin bank account number - the best I can do is to send you a long list of 32-bit or whatever BIP32 chain of addresses. Just don't mix them up, transpose any digits, or mislay them.
The banking system doesn't tolerate mistakes in their account numbers either.
And let me know when they run out so I can then send you some more!
It's essentially an infinite chain, so when infinity runs out, you will indeed have to furnish people using it with a new one.
The LN proposal with direct channels is interesting, but not sure its exactly what I have in mind.