I noticed that for the Google Play Services you mention that it is for web only meaning we have to host our game in server. Actually It could work for Chrome App also. But you need to modify the permission in manifest.json and get the Chrome Identity and put it in there also. Hopefully when exported to Chrome Web App, you can includes some fields for us to key in the Chrome App Identity thing. Or maybe in the plugin object property?.
This is the sample link (noticed that there is a guide on how to use the services for Chrome Package App )
This is the link on how to get the Chrome App Identity to include in the manifest.json
I created this post since when I tried the google play services for Chrome App, I noticed that it does not include Chrome Identity permission to work with Google Play services.
I hope this helps.