I know this is an older question but felt another answer was needed.
Running a mailserver properly is hard work these days, with staying off blacklists, IP-reputation, DKIM, SPF, etc. etc. It's tough to get your mails into the inboxes of providers like Gmail, Hotmail, Yahoo, etc.
If your sending from each server directly you have to take care of all that, for each server separately. Instead, you can relay all your outgoing mail through an SMTP relay. Gmail is one option. Services like https://sendgrid.com/ and https://mandrill.com/ are more specialized in this. They usually offer a free tier that may suffice depending on the amount of mail
you send and aren't harder to set up than gmail's smtp.
A benefit of a specialized mail relays is the extra information they provide. I get a centralized dashboard of mail traffic from all my servers so I don't need to gather Postfix logs from the individual servers. I can quickly see mail volume, delivery problems, search for individual mails or find patterns, etc. and the service will handle smtp reputation for me to prevent delivery problems. In my experience, delivery rates went up while I spent less time on it.