SSH private key, used for authentication or encryption?

I’m confused around the concept of SSH authentication. Is ssh key excange necessary for actually authenticating or just encrypting the login attempt and subsequent data exchange?? I mean isn’t username/password used for authenticating?

In other words, if someone finds my user name and password by brute forcing would he be able to login without having hold of my client private key?

And why use a private* client* key instead of a public client key with a server private key?