How to automatically deliver /var/mail/USER to my Gmail?
On my High Sierra (10.13.4) machine, mail honors the .forward file. Create a file in your home directory called ".forward" containing an email address. All your mail on the Mac system will be forwarded to this address.

It makes good sense to limit access to this file, with chmod.

For example (open Terminal in your home directory):

cube:~ borg$ cat &gt; .forward
locutus@borg.org &lt;return&gt;
&lt;ctrl-d&gt;
cube:~ borg$ chmod 600 .forward


Then you can test by sending a mail to yourself, from the command line:

cube:~ borg$ mail borg
Subject: test
This is a test.
&lt;ctrl-d&gt;EOT
cube:~ borg$ mail
No mail for borg
cube:~ borg$ 


Note: this only works for forwarding MacOS system mail, not for forwarding from any of your cloud mail services. You can enter any valid e-mail address in the file.