Neither Microsoft Office nor Google Docs "just worked" on tablets and phones without major reworking of their UIs. It's not obvious that the web has any advantage here - I just now visited docs.google.com and was encouraged to "download the Google Drive iOS app" so that I can "edit documents." (!)
If you target multiple platforms with one app, you get an app that works from OK to poorly on lots of platforms. Look at Light Table as an example: done entirely via web technologies, easy to port everywhere, but feels incredibly alien and _wrong_ on my Mac. (No offense Chris!)
Compare to Sublime Text 2, which to my understanding has lots of platform-specific code to make it feel native on each platform, but also a shared core (using Cairo, etc). So "six native apps" need not cost anywhere near six times a single native app.
So in the end, the cost of targeting multiple platforms with a single app is surely lower than targeting each platform individually, but that's just a classic cost/quality tradeoff - the web limits your polish. And what good is having your app on multiple platforms if it's inferior to native alternatives on all of them? I know as a Mac user, I'll pick the Mac app that feels like a Mac app every time.