If that was really true, all those "browser-only" appliances would have taken off. Redmond and other tech co's have tried to migrate their tools to browsers and so for it hasn't worked. Rumors about Google OS have been circulating.
Most organizations have a suite of productivity, engineering, financial, graphics design, and yes....browsers.
Operating systems DO matter!
Where have you been for the past 6 years?! MicroSoft Word and Excel already run inside IE6. Adobe's Acrobat runs on just about every browser, too.
None of those on-line aps care about your OS.
Caring about the OS is old school. It's the browser that matters now and in the future...because people spend most of their time on-line rather than in stand-alone programs.