Dovecot SMTPD Sasl auth error: fatal: no SASL authentication mechanisms

As you can see, there's no smtpd_sasl_path in our output of postconf -n

Why? The answer is easy. Because there's no smtpd_sasl_path set. Have a look at your first snippet: There's a typo in it. You wrote smptd_sasl_path = private/auth -- but there's a difference between smtpd and smptd.

Because Postfix doesn't have informations where to find the socket, Postfix is claiming the "failing connect".

Fix it and it'll work. :-)

And, at the end: Remember for the next time: Reading postconf -n carefully and make sure, that Postfix has the same information as you believe Postfix should have, is always an important step to find the failure.