Would think one needs to get the web interface working first before testing with the moodle app.
Yes, it's a little different ... there are 3 tabs across the set up of Google IAM. All three must be completed. You mentioned terms of service and privacy statement. I have made those static web pages (.html) and use the same ones for the sandbox sites (3.5->310) all of them have Google working. I cheated there and 'borrowed' a tos.html and privacy.html page from a organization for education ... changing the appropriate names/references in the borrowed pages to mine.
One of the tabs has to do with verifying that you own the server/domain via DNS. They offer adding a txt record to your DNS for the moodle or, perhaps easier and don't have to include DNS server admin, is the option to verfiy ownership via an HTML file at the root of the site ... ie, moodle code.
They don't provide the HTML file ... just describe ... like name it with 'stringname.html' and contained therein a one liner with what appears to be the same verification string.
The other thing I do which in docs say isn't necessary, I have a Google System account to use with IAM. In the checks for setup, 'System Account Connected' ... which has to be the same one used when first setup.
In versions of Moodle <-310 I have such a system account. On a 310 I don't have that setup and it works anyway.
Do you get the login screen that comes from Google? (that shows something like the following)
The SOSSIG in above screen clip is the name of the IAM I am using.
Can PM you some links to these sandbox sites so you can try them out just to see how they work. I'll remove your test accounts after you let me know. ;)
As far as keeping docs up to date ... one can edit them. Will say that I'm not gonna volunteer to do that ... besides ... Google has been known to change things. :| I already have a job and enough to do.