I made some research on the subject. And I am not alone having this kind of difficulty. It seems that Discourse is incompatible with some SMTP servers or at least, some configurations.
On StackOverflow, SMTP Error <<: 503 AUTH command used when not advertised, David Spector is referencing this blog post: Tuning cPanel TLS for SMTP incoming email reception.
It might be that Discourse is trying to authenticate before using SSL or STARTTLS. Is this a security issue? Is Discourse needing something to fix or is it a problem related to SMTP configuration? It is not clear to me.
Please share your thoughts.