Maybe it isn't the device, but it is the email provider? Gmail (Google) dropped "push" support for all but their own devices!
I say push back at Google! Find a new email provider who supports what you need!!! People talk about "Apple Sheep", well this seems to me to be a case of "Google Sheep" !
I've said it before: if your local pizza shop stopped delivering, would you go get it or find another pizza shop that DOES deliver???
I've been using Gmail since it launched as a Beta service all those years ago but as much as I am prepared to use workarounds such as Mailbox or the Gmail app, push e-mail is vital to me. Over the last month I have been testing with an Outlook.com account, it gives me push and everything else I need so I might just have to use Outlook as my primary account from now on.
Mailbox is excellent and is an excellent workaround but it's still a third party app at the end of the day. The Gmail app is a bit of a calamity and Google infuriated millions of Gmail users with stupid and annoying UI changes over the last few months. Having said that, I tried to upgrade my existing Google account to a Google Apps for Business account but you cannot upgrade a Gmail account. So even though I was prepared to pay whatever it took to get EAS back Google roadblocks every avenue.
I honestly, honestly hope I can get EAS working with my Gmail account on my new iPhone when it arrives, but if not I will just have to migrate to Outlook. At least Microsoft can give me what I require.
Like you so rightly say, you should go with the service provider who provides what you need.