Dovecot / Postfix / SASL: Connect to /var/run/dovecot/auth-client failed: Permission denied

You define this parameter in

smtpd_sasl_path = /var/run/dovecot/auth-client

This tells postfix where path the unix socket for SASL connection.

Yet, in dovecot config you define the SASL socket was in /var/spool/postfix/private/auth

service auth {
  unix_listener /var/spool/postfix/private/auth {
    group = postfix
    mode = 0660
    user = postfix

Try to set smtpd_sasl_path = private/auth and restart postfix.