Why can't you? Well, for one, social media companies don't want you to save stuff locally, because they can't serve ads with local content. Furthermore, browser APIs have never embraced the file system because there is still a large group of techies who think the browser should be for browsing documents and not virtualizing apps (spoiler: this argument is dead and nobody will ever go back to native apps again). Finally, the file system paradigm fails with shared content; you can't save a Google Doc to disk because then how can your friends or coworkers update it? It's much easier for Google to store the data on their server so that everyone can access it instead of you setting up some god-awful FTP-or-whatever solution so that your wife can pull up the grocery list at the store.
I'm hoping the new Chrome file system API will bring a new era of Web apps that respect the file system and allow you to e.g. load and save documents off your disk. However, this still won't be good enough for multiplayer apps, where many devices need to access the same content at the same time. I don't know if there is any real way we can go back to the P2P paradigm without destroying NAT - WebRTC tries but WebRTC itself resorts to server-based communication (TURN) when STUN fails.