Shouldn’t web apps behave like real apps?
That is a goal of Progressive Web Apps (PWA). Google, Microsoft, maybe Apple and Mozilla and some other companies are working on a new, modern web application standard so web apps act more like native apps. (Apple is adding PWA features to the Safari browser.)
They are still hosted directly on a website, but the advantage is that developers can update them directly on their web server without submitting those updates to several different app stores. And the same app should run on all browsers and platforms.
The “progressive” part means they are enhanced with modern web features. They would work in older browsers that don’t support the new features, but will work better and with more features in modern browsers. They will get their own window and shortcut on your taskbar (on Windows 10 and anything running Chrome) or an icon on your home screen (on Android devices and other smartphones).
Unlike Google’s Chrome Packaged Apps or Microsoft’s Hosted Web Apps, these do not have to be packaged as a file and submitted to an app store. But PWAs can be listed in app stores for easier discovery and installation, though the store will just point to the web app’s servers.