I've been waiting for a decent iOS Gmail client for so long. This is an excellent first release, far better than Google's own half-hearted attempt. Let's hope they get some traction petitioning for an acceptable method to send push notifications: https://news.ycombinator.com/item?id=3706993
I think the issue here is that it isn't really "push" they want to do. From what I can tell, they want to do is simply "run all the time so we can keep a connection open to receive new messages immediately". Which, to be fair, is pretty similar to what Apple does for their push notifications. However, allowing this would be a bad move. It'd mean that rather than once centralized connection that knows about the device…
There's nothing inherently wrong with this on mobile devices. If Apple can't do it for iOS, that is a flaw in their design, or their thinking.