What is the role of leaf certificate's private key if this certificate is signed by the intermediate certificate's private key and the certificate signature will only be validated using its public key.
See The First Few Milliseconds of an HTTPS Connection, by Jeff Moser. Scroll down to the section, 'Trading Secrets'. This section describes how the client encrypts the pre-master secret using the server's public key (i.e. the public key contained in the server's leaf certificate), and sends the encrypted public key to the server. The server then has to decrypt this message. This is done using the server's private key corresponding to the public key in the server's leaf certificate.