Why does a SSH server need a host key-pair?

You already quite understand how the public key cryptography works and how the client is authenticated to the server using public key authentication.

The same thing is needed on the other direction. Internet and computer networks are evil place and it is quite easy to redirect traffic, spoof DNS or somehow make you connect to evil host, which would like to capture your password or other sensitive information. And it can not be ensured using public key only, because everyone can send you this public key, but only the server you want to connect can send you signature of your data that you can verify using the public key of the host.