Adding the shortcut to your home screen pretty much treats it like an app (loading it in it’s own container instead of has a page under the chrome/other web browser).
I think it’s a little silly. The whole point of having a responsive website is so you wouldn’t have to create a whole separate app. That is why the whole “add shortcut to homescreen for websites in order to emulate an app” is a thing. A lot of people just forget this option even exists.
The only area I think an app is really necessary is when you have a fire tablet or other device that doesn’t let you place shortcuts on the home screen for websites. Then an app would be awesome.
EDIT: I guess if you want to do push notifications an app could be nice. With how many third party apps there are for this it is probably unnecessary though (I don’t see a benefit of using HA for push notifications over Matrix, Pushbullet, or any of the other services I use).