This one is actually on Google, there is no way to "push" Gmail to an email client (apart from their own) because they don't fully implement the protocol. They used to support Microsoft Exchange, but they dropped it for free accounts. Third party apps go around this by storing your credentials/login session on their own servers and check your emails every minute by hitting the Gmail endpoint, and send a push to your phone. Obviously this is something Apple would never do.
If you prefer using Apple's own email app instead of Gmail, you can forward your Gmail to an Outlook or iCloud email. Those do support push to iOS email app. Then change your SMTP settings to send emails through Gmail.
Ahhh right so not an Apple flaw but a Google one! I can drop that from one of my few Apple bugbears since switching over then.
I do love this phone, it just does all that I want and feels quality. Most normal use days I end the day with 30-40% - if I've hammered it then 20%ish and if it's been a slow day 50-60%. I've never had it close to dying and that is the best thing about it.