Curious about this. I have 5 email accounts on my iPhone but only gmail and iCloud set to push. Push is working for me on both accounts.
My gmail used to be set to push as well but this is no longer an option for me. I think I lost it when I upgraded to a newer phone. Maybe they let people who currently had it set up keep it but stopped allowing people to set it up to push going forward.
When I look at my mail settings in iOS the "fetch new data" setting is set to push. If I tap on that setting it shows that this is only true for apps that use iCloud though. My gmail account only has "fetch" or "manual" as selectable choices. Mine is set to fetch every 15 minutes which is plenty for me. If the message is urgent enough that it can't wait 15 minutes they can text or call me.