Most people don't even know they can use extensions (or even what a browser is), but they do know that Facebook has a chat function, because it's right there.This seems to suggest a web app or suite of web apps that "competes" with Fb, in some sense. You don't have to install (or even know the URL of) a web app! Mozilla may get there eventually, but they're not going to travel the obvious, NSA-(or plug in your favorite bad actor-)vulnerable way. I think their current work with identity on Persona could grow into that, but they're going to keep it distributed, which I appreciate.
EDIT: Now that I've RTFA, I find that you also are thinking along these lines. However, I think you've glossed over who is actually running the server. In order for Persona to function, in order for all my data to be synced across all my devices, in order for any of my friends to see any of my comments, someone has to pay an electric bill every month. If there is a vast marketplace of server-operators, they can compete on price, {NSA, advertising}-resistance, etc.
I could see Mozilla carrying the web to that paradise of individual autonomy, but I could also see Fb and Fb-alikes derailing the effort along the way. They embraced-and-extended the web once already; I can imagine them doing it again: "Sure I can chat with my friends and "like" my favorite brands through Firefox, but did you see that Facebook allows me to frobnicate my Facebook snorrzlers as well?"