I waited 2 hours for an email yesterday only to find it hidden in a new "Promotions" tab instead of my inbox, which meant it didn't get pushed to my phone for some reason.
Man, waiting on Steam Guard to send me an email with a confirmation code got pretty hairy during some of the recent sales. When there's 15 minutes left to say 75% on a game I really want and the confirmation email gets sent to Promotions and therefore never seen... thanks Gmail.
Spam made me give up running my own mail server and start using Gmail.
Gmail is slowly making me give up email altogether.
I am not saying that email perfect and doesn't need innovation, but there should be greater sensitivity towards complexity and friction on Google's part. It is bad enough already that various functions are spread semi-randomly around in different buttons, drop-downs and links so you have to click around to get things done. There's really no need to try to get too clever about things.
I'd like more predictable behavior ("where did that email I looked at just 5 minutes ago end up?") and I would really like a better filtering system. They should have a look at the scoring system in Gnus and then think long and hard about how you build a sensible UI atop that kind of expressive power.
For the iOS gmail app you can choose to be notified for all new mail or "primary" only. I'm not sure about using a built-in client like the Mail app though.