Yes, that might work for notifications from moodle to teachers/students/admin, but if one decides to force subscribe all users of a moodle whose total user base is, oh, let's say 5000 users, Google might not work for that volume of email traffic.
You mentioned settings from the Google end ... mind sharing what they were called and their function? (not your login/pass/URL).
If you plan to use a lot of forums in your Moodle, might be better off using and SMTP service that is designed for volumes of traffic beyond what Google will allow.
One such service is SendGrid ... and their API Key setup. Moodle can be configured to use SendGrid and the API key setup.
Please see:
https://moodle.org/mod/forum/discuss.php?d=392578#p1662786
Strongly suggest, that now you have Google working, and a test message has been received, look at the full header of that message to learn now the path email takes from your moodle server to whatever inbox. That might prove to come in handy in the future.
'SoS', Ken (a former email/email gateway/listserv server administrator)