Maybe I'm old school, but this idea of archiving mail instead of deleting it just doesn't sit well with me, I have over 22k emails in one inbox from god knows how many years, and I do back this up periodically, which takes ages! I don't want it clogging up with crap mail I've hit the delete key on.
Now Google Mail (GMail) when you send the delete command via Mac Mail using their recommended settings of not moving to the trash box and not leaving the message on the server simply archives the message. This is a pain.
If you want it to truly be deleted (i.e. moved to GMail's trash) then tick both of those boxes under Preferences > Accounts in MacMail then click your GMails trash folder in Mac Mail, click Mailbox > Use this mailbox for > Trash.
Now all mail you delete will go to the GMail trash and GMail will automatically delete it after 30 days.