Allowing apps to access browser cookies is a solution to the problem that apps and websites are disconnected and don't know about each other's state. That's a different aspect than the one you're talking about.

Yes, you can build responsive sites, but what you can do in the browser is limited, especially when it comes to flaky connections that are still very typical. To take the example in the post a little bit further: Imagine you're on the train and are reading a multi-page New York Times article. Train goes into a tunnel. Trying to load the next page will result in a connection error. Native apps can know about your network state and can pre-cache content you're about to read next.