> no need to exchange the key during authentication

The public key does need to be exchanged, along with a signature relating it to the current session. This is all handled by TLS, there is no need for the client to send the key in the application data.

> You still need to generate it and distribute it to the user

This approach avoids distributing secret key material at all. Private keys should ideally never move. They are generated randomly, used to derive the corresponding public key, and then persisted as appropriate. The public key is sent around to other parties.