Doesn't the Gmail app on iOS use a push-system for delivery? Why would you want to use EAS for Gmail? Seems reasonably (and positively Apple-like) for Google to push people into the Gmail app where they control the experience and can surface first-party Gmail paradigms more readily.
Admittedly I'm an Android user with limit experience with stock iOS mail app. Is the iOS mail app really that much better than the Gmail app?
Gmail.app for iOS is incredibly sluggish (on my 5S) while Android's version on my Moto G is pretty snappy, so you're not experiencing the same thing at all.
I run both Mail.app and Gmail.app on my iPhone because I'll often get the push notification for Gmail.app, tap the notification to open the email, get frustrated and tired waiting for it to load and just go manually refresh Mail.app and see it almost instantly.
Admittedly I'm an Android user with limit experience with stock iOS mail app. Is the iOS mail app really that much better than the Gmail app?