"The next battle in the war for dominance over the Web is going to be waged at the browser level." -> Nah. This has been tried before, what's different now that would make it suddenly work?
Google's plan with G+ isn't to overtake Facebook. It's to commoditize social.
The Google+ App, like the other Google ecosystem apps are just linked to the website. Aside from that, the other extensions only really compete with Facebook to the extent that Facebook extensions exist. I believe Facebook might build an OS in the future, but reception might take a while for those security conscious individuals who'll be hesitant to adopt a browser that presumably harvests more data for the Facebook machine.