When setting-up my mobile email client, I usually need to enter the SMTP relay server domain name, but why my email client need a SMTP relay server at all?
I think it can directly talk to the smtpd server (e.g. postfix) from which domain the destination Email address is, by looking up the DNS MX records. And send directly to it, why not?
Actually I have another confusion, if a smtpd server use SMTP AUTH, then how can a rely server elsewhere rely a mail to it? The rely server simply don't have the username and password necessary to transfer that mail.
Email clients use SMTP relay server to
Direct email sending might be a reasonably simple option in ancient pre spam past.
SMTP AUTH is used (mainly) with email client to email server communication. SMTP server to SMTP server communication could not require SMTP AUTH without prohibiting email from "strangers". SMTP AUTH allows to accept messages from trusted clients (customers) even from IP addresses that otherwise would be blocked by DNSBL (e.g. DUL ranges).