I have been using Push Mail for my Google Apps account for many years on my iPhone and iPad using the native mail application.
You need to add your email account as Exchange account and not Gmail. Everything else you would have having except in that you should use m.google.com as server. The domain field can be left blank as it is not mandatory.
You get the mails instantly pushed, you can see the real Push feature using this setting. You have the option to even have push setting to particular labels too.
Gmail users rejoice as iOS 11 may have a solution for the long-standing push notification woes. No longer having to rely on a third-party app, users with Google email accounts should be able to jump back onto iOS’ native Mail app.