That explanation doesn't really make sense to me.
If something could be built as a native app without depending on a central server, it could also be built as a PWA without a central server. You don't need to store user data centrally at all, just because it's a webapp. You can just have the clients use localStorage or IndexedDB or whatever.
You still have to host the static files for the webapp itself, but that can be made very cheap.
Of course, API feature parity between native and web apps is a separate issue. But the argument about server costs doesn't seem like a good one.
Isn't localStorage limited to 5 MB of data?
Sure, but localStorage isn't really ideal for storing large objects anyway, because it forces everything to be stored in one big string-to-string map. It's great for small amounts of data such as user preferences.
There are other APIs that allow you to store binary data directly (which you'll probably want if you're storing large files) and also to use/request larger quotas.
IndexedDB API is a bit more liberal in that regard
Yeah, better is the filesystem API