How to forward email to GMAIL using EXIM4

Welcome to the world of big email players in general, and Gmail in particular.

Gmail is blocking mail based on "reputation" in order to fight spam. That is, it only accepts mail from servers that are known to be well-behaved according to some metric they set unilaterally. It is nearly impossible for a single new server outside an established mail organization to build up the necessary reputation to be able to deliver mail to Gmail. (Other big mail service providers act similarly, although most not quite as recklessly.)

The recommended solution is to not even try to get accepted as a real mail server, but instead send your mail as a client would, via message submission to port 587 of your mail provider with SMTP AUTH using the credentials of your mail account.

By the way, this is not specific to Exim. The exact same thing would happen if you tried doing it with Postfix, Sendmail, or Microsoft Exchange.