Without them, I would have to regularly check the app of interest for any important updates, losing both time and mental space with another thing to keep in the back of my mind.
Thanks to push notifications, I can defer interacting with the app to when it's actually relevant.
Also as I argued in another comment [1], web notifications are simply better than native app ones. Push notifications are here to stay, the choice is in the implementation, not whether they'll become culturally accepted or not (they already are, iOS users simple are nagged to install native apps to have them).