To put it another way, the benefit I get by having my computer verify that I'm connecting to news.ycombinator.com and not some fraudulent website is so that I don't inadvertently transmit my account credentials to a scammer.
The benefit news.ycombinator.com gets by verifying a client certificate I send them is that they can verify that the user u801e is establishing a connection to them and not some scammer who happens to have my account credentials.
Those benefits are lost if either the server or client sends a self signed certificate, because there's no one who we can use to verify the authenticity of the certificate. The reason using an external CA is not needed on the server side is, in my opinion, because the server is the one who's handling the account creation, so it doesn't need an external authority other than itself to verify whether the certificate is valid. If the server just accepted a self-signed certificate, then how does it verify that the certificate is valid?