It wouldn't be a "new" way to badger users, because developers can currently badger users to install a native app. A Smart App Banner for installable web apps would be no worse than what Apple currently offers, a Smart App Banner exclusively for native apps, and I think those banners are just fine.
https://developer.apple.com/documentation/webkit/promoting_a...
> Smart App Banners vastly improve users’ browsing experience compared to other promotional methods. In iOS, Smart App Banners provide a consistent look and feel that users come to recognize. They trust that tapping the banner will take them to the App Store and not a third-party advertisement. They appreciate unobtrusive banners at the top of a webpage, instead of a full screen that interrupts their experience with the web content. And with a large and prominent Close button, a banner is easy to dismiss. When the user returns to the webpage, the banner doesn’t reappear.
Note that "Add to Dock" installable web apps already do have a Smart App Banner on the new macOS 14 Sonoma, but only when you already have the web app installed. (It's exactly like the Smart App Banner for native apps, but, for web apps, it only works with the "Open" button, not with the "Install" button.)