For iOS, can't you just wrap your web application in a thin native app layer that provides the push functionality?

From the app store guidelines:

4.2 Minimum Functionality

Your app should include features, content, and UI that elevate it beyond a repackaged website. If your app is not particularly useful, unique, or “app-like,” it doesn’t belong on the App Store.

4.3 We'll continue to get away with this until the EU makes us stop.

Would push notification support not be exactly the kind of feature that would qualify for this rule? In practice there are hundreds of apps like this on the app store.

Right! I ran in to this when I wanted our app to “portal” to the website and just move Bluetooth data along. Apple thought about this and doesn’t like it.

Perhaps the main distinguishing app-like thing is distracting notifications.

Yes you can and there tons of apps that do just that.

The important part is that they go through the app store, so that apple has complete control of what goes through.

Yes you can and we are exactly doing this since quite a while. Actually a lot of companies do this.

You can, you just can't publish it on the appstore.

How do you distribute it in that case? Outside of enterprise apps, which are not allowed to be public.

Sorry but I think I don't understand your question.

If you use a native wrapper and load a webview inside you can publish it in the AppStore.

They probably embedded the browser in a client.

for iOS you can have a work around. The users using the web based app in safari, then whenever a notification needs to be sent, twilio api will send a SMS message to alert the user. Of course it will only send an sms message if the user allows it and enters the mobile number.

Short answer: He didn't. You cannot do it without your app being in the App Store.