+1 to Just H's response.
Have several setup and can provide URL's to you via PM if you want a look see - a 2.8, 2.9, and a 3.x providing and encouraging use of Google to authenticate. It does change perspectives in how one does things in Moodle. Some of it better ... some of it not so good. Student will have EMail addresses that are part of the GAFE for the entity so restrict EMail addresses to those domains + the domain used by teachers.
One thing to know ... while students might be using their Google credentials to login to your site, the plugin isn't 'single sign on' and it does have some behaviors that you'd need to investigate IF students were given shared Chromebooks.
Best if your Moodle running https cause Google really doesn't like to talk to http boxen ... just like Moodle when it comes to using the built in install/updates to addons/plugins.
Tend to agree with you in your comparison of Google For Education (Google Classroom) ... which is like Google docs/sheets on macro steriods. They are adding new features all the time and think they too will eventually reach the same issues Moodle has when it comes to core vs plugins, etc.. - 'feature creep' which adds to complexity.
I do hope, Moodle HQ and the Association as thinking about greater/better integration right out of the shrink wrap cause if non are looking over shoulder, even Microsoft with o365/etc. and their online suite has a 'classroom' addition now.
Anyway, your Moodle is NOT done yet ... but you will have to start now in installing and testing the authentication addon suggested. You will be questioned by your peers and admin ... more than likely, and it takes some courage to stand alone.
TIPS: provide a 'Google LOGOUT' button/link in custom menus of your Moodle. Logging out of Moodle doesn't inform Google user has logged out. Important on shared machines/devices as Moodle's session timeouts are quite long (and shouldn't be reduced, IMHO).
And use the custom drop down menus for things related to Google but point them to your GAFE domain.
Don't think they will mind ... see http://moodle.wisd.net/moodle2/
check out the drop down menus ... site, BTW, is restricted to valid wisd.net domains.
'spirit of sharing', Ken