Push notifications from the GMail app and Mailbox are not the same as what you get with Mail.app. With Mail.app, mail is downloaded in the background and immediately available when you launch the app. With the others, you launch the app and then stare at the screen for 5 - 10 seconds while mail is downloaded.

iOS 7 is adding a feature that will allow downloading data in the background and hopefully Mailbox and the GMail app will take advantage of that. But in the meantime: fetch, get Google Apps for $5/mo, or deal with the annoying wait

Yes, but it doesn't let you combine multiple disparate (not all Gmail) inboxes into a single pane of glass like Mail.app does.

I found the gmail iphone app operates very poorly with a slow or spotty internet connection. It seems that google assume everyone is connected all the time with a fast connection. With the ios mail app it can download mail in the background slowly when it can and I'm able to read it offline.

My default reply to these comments is "embrace, extend, extinguish".

P.S. Ironic that your nickname is "gnus" :) (http://en.wikipedia.org/wiki/Gnus)

I agree that the push functionality seems to instantly notify you about incoming mail, however it doesn't seem to integrate as well with iOS. Have you been able to get email forwarding in other apps to use the Gmail app rather than trying to use Mail?