I've definitely removed websites from sending Push notifications, but I can't think of a single time I thought: "Oh I really wish this site had Push Notifications".
IMO, online services have been using email as the default form of notifications for a long time, people like emails because:
- they can set up sorting rules
- they get notifications on all devices, and if read, is marked read everywhere
- they can check it on their own time
With push notifications you are:
- displaying notifications a way where their only choice to limit seeing them is to remove them entirely
- only notifying them on one device (hopefully) and not sending the same unread Push notifications to every device even if you've read it on one :/
- interrupting the user during the middle of whatever they're doing
I'd love to see a case made that web push notifications could ever be useful, but I haven't seen or heard that yet.