Answer is straigtforward but not clear from the docs:
1 - Do not use the optional "Hosted Domain" input
2 - In your Google API console you also have to (and this isn't self evident, or even present) from Google's Documentation):
- Go to the "Library" (aka link in the left-hand sidebar that enumerates available APIs)
- Click on "Google + API"
- Near the top of the "Google + API" display click "Enable"
It should now work.
We have tested this on two separate flarum installs. One on AWS/EC2 and one on a local, but publically routable KVM/Ubuntu setup.
Many thanks to our intern Moh who figured out this tiny crucial step. Lord knows I couldn't.