I agree about the drawback of lowering security. The dealbreaker is my imaginary conversation with my client, telling them to lower their gmail security. It's not a solution.
This solution at least got me to the point where my message sent successfully, but the message still isn't showing up in my gmail account. But maybe it'll work for others.
https://wordpress.org/support/topic/contact-form-7-godaddy-gmail?replies=3
this is informative as well:
http://kb.cf7skins.com/contact-form-7-email-issues/#hosting