I'd say the one really important thing web is missing is notifications for when your not on the site. I don't really know how a browser could ever implement in a similar fashion.
Email, or SMS? Though I don't see why Apples Push Notification Service, can't be opened up for the web side. They are already going through Apple servers correct? Safari has push notifications, let's combine the two.
http://w3c.github.io/push-api/
https://developers.google.com/web/updates/2015/03/push-notif...