Maybe they should be "the browser is the Operating Environment" instead of "the browser is the Operating System". That would be more accurate.
Essentially what this shift is doing is it is changing the way developers build applications for use by the masses. Instead of coding back-end processes to a certain file-system and the front-end GUI in the form of modular windows for a particular OS, developers are increasingly designing their GUI's to be rendered in a browser and building the back-end on remote servers.
Google (and others) are closing this gap a bit by providing the same functionality remote servers provide, and implementing this functionality on the individual client machines (Gears)... not necessarily as a replacement, but as a fall-back when Internet access is not available or is shaky.
This design philosophy, coupled with the fact that the front-end GUI is also hosted remotely, is basically making the browser the access point to the applications people use on a daily basis... whereas, before, the user's desktop (the visible portions of the operating system and the reason most people upgraded from one operating system to the next) was the access point to the applications people used to use on a daily basis.
The shift is still happening, and it won't happen overnight, but more and more people are turning to their web browser for every task, rather than the Start Menu in Windows, for instance. Rather than running Calc to do a calculation, they might have a Calculator Gadget on their customized Google Home Page (or other equivalent site) or maybe just typing the numbers directly into the Google search box. Rather than running Solitaire on their local machine, they might be visiting a website to run a Flash-based version of Solitaire. Rather than opening Notepad to type some notes, they might be using a hosted document provider. This trend in shifting from the general desktop to the browser is continuing to happen.
While it is true that it isn't necessarily replacing the "OS" from a geek standpoint (after all, the OS is still needed to RUN the web browser), it IS replacing the public's perception of what an OS is. Most people who just USE computers and don't also TINKER with computers generally move from one operating system to the next because of the next wave of applications that this opens up to them... because of the interface changes that make doing every-day things easier... etc... Nowadays, people are staring less at their desktops and are staring at a web browser. So now, people are more interested about the advancements in web browsers than the advancements in operating systems.
Eventually, the operating system will be a moot point. One day, a Windows user will have no problem switching to Linux... because he/she won't have to worry about "will Microsoft Word run on Linux? Will I have to get used to a whole new desktop experience?" and so on... instead, the only question will be "Can I run a web browser in Linux? If so, will all of my web-based applications run in that web browser?" The answer to both of those questions is usually already "yes"... so the majority of the concerns are already taken care of.
The main areas that will not consider this kind of switch are Enterprises (which have legacy applications which will continue to remain in their respective environments) or hard-core gamers (who rely heavily on installed applications and powerful hardware configurations to meet their hard-core gaming needs.) The latter group might be eventually replaced by the console market as the line between "what is a gaming console" and "what is a computer" is continually blurred. It would not surprise me if the next PlayStation comes pre-installed with Linux, with much easier use by laymen... and if the next XBox 360 comes pre-installed with the latest flavor of Windows, complete with tie-ins with Microsoft's cloud-based services. Add a keyboard and a mouse and a hi-def television, and you're set.
As for the Enterprise... it is only a matter of time before legacy applications catch-up to today's technology shifts. A lot of applications which once ran on mainframes are making the shift to the Windows environment... and then will eventually make the shift to the hosted application paradigm. In the meantime, Microsoft and Google are already making headway in the lesser applications like word processing, spreadsheets, etc...