The extension provides the majority of the service’s functionality; adding papers to your library, sourcing the correct reference information, sorting your library by label/folder, and so on.
However, it isn’t explicitly required if you only want to read PDFs that are already in your library. Those are just stored in a folder of your Google Drive account. That means they can be opened just like any other PDF, using either the local GDrive app or the web interface at https://drive.google.com
You won’t be able to use all the metadata for sorting the library, but Google has pretty good search functionality within Drive - including the full-text search.
Your highlights and comments are embedded in the PDF too, so anything you do outside of Paperpile’s extension should be visible within it once you get back to your personal computer.