Google's apps for iOS are subpar compared to its Android counterparts, or even to other iOS apps. I'm a heavy Android user, but on my new iPhone I actually prefer the native email app, because it is simple, effective, more standards complaint than Gmail and has quick swipe actions for both delete and archive, which is all I need.
On push, Google Apps gives you ActiveSync as a protocol, as an alternative to IMAP. But the integration with iOS Mail is weird and I could not use it.
But FastMail on the other hand, in a twisted turn of events supports push through Apple's own APN, getting the same treatment as iCloud: https://blog.fastmail.com/2015/07/17/push-email-now-availabl...
FastMail also has a native app that does push notifications, but it's just the packaged web interacts with push notifications added. This is both good and bad. It's good because the mobile web interface is very decent, compared to Gmail, and this means you always have a decent UI on whatever OS you have. It's bad because it doesn't feel native, but then there's nothing more native than the iOS Mail app.