Not true - they could check email and send you push notifications server-side, but still have the client talk directly to gmail for nearly all functions.

> they could check email

That requires non-google action. That is a point of failure.

Sure, but the failure means that you don't get push notifications - you can still use your email app.

That is correct, and has nothing to do with my original statement:

No second point of failure => no native push notifications.