The headsets don’t really make sense to me in the way you’re describing. Phones are omnipresent because it’s a thing you always just have on you. Headsets are large enough that it’s a conscious choice to bring it; they’re closer to a laptop than a phone.
Also, the web interface is like right there staring at them. Any device with a browser can access Facebook like that. Google/Apple/Microsoft can’t mess with that much without causing a huge scene and probably massive antitrust backlash.