The fundamental problem with this idea is that the computer has to store your password somewhere, for it to be able to check it. Now, these particular passwords could be kept only on the server, but they are still kept. In a file. On a computer.

Your comment comes across as somewhat silly and akin to saying that the problem is we have to transmit the passwords (or some derivative data) across the internet to do business with someone. Yes... that is a facet of the problem. It is also a facet of how work gets done.