On iOS you can't get notifications if you've closed out of Safari... it's not really usable.
A simple scenario is - you're inviting someone to a chat room. You send them an invite and now you have to wait. Usually what people do is other things - like go to a different website, go to a different app, etc. In only the case where you're sitting on the same website will you get a notification. It doesn't work.
It gets worse - let's say you installed the app to your homescreen. You run through that same scenario above - you navigate through your web app into /chatroom and click on an invite button there. Then you navigate away to another website/app. When you come back to the homescreen app, the entire site has reloaded and it will take you automatically to the main screen of your web app: /
The progressive path isn't great, but installing an app from an App Store is a huge jump for users to make, so neither is ideal really...