As an example, I refuse to install a native Facebook application because of privacy concerns, but a web version with push notification support could provide the same functionality.
- Instant messaging notifications (e.g. Riot.im, Glowing Bear, WhatsApp, Discord, Slack)
- Changes to files in cloud/collaboration services
Basically, any time you would use a push notification on a mobile device, but when the "desktop application" is essentially just an electron wrapper of the web UI.
Just one example of legit use not for ads.
I also run Microsoft Teams in the browser, getting notifications for messages is quite useful.
Forums are an example where the website is perfectly fine without an app, but I'd really like to be able to choose whether or not I get push notifications from them on my phone.