Mail app?

The only advantage I can think of is that it'd work with new iOS 9 features like "Siri, remind me about this" as that won't work with third party apps yet (as far as I know).

I've posted (as have others) that there is seemingly something going on with using the stock mail app with gmail addresses - as it, seemingly, doesn't stick to the Fetch schedule sometimes. For example, an email will be sent to me at 5:08am, but might not hit my inbox until I actually open the app at 7am or later, although I've got Fetch set to 15 minutes. This has been happening on iOS 9, 9.0.1, 9.0.2 and now 9.1.

Using an app like google inbox would be nice because I'm fairly sure some of those third party options support push. I just can't lose the totally integrated feel of the stock app. Also, I can only comment on gmail account as I haven't used other services (although, iCloud sounds like it would be a nice switch - thoughts anyone?)

Hope this helped at least a bit.