About the same time the Web was spinning up, network bandwidth and computer performance both started to drop in price, giving software designers the option of retaining a portion of the processing on their end. While it does not replace the standalone desktop app, a "cloud" app affords many advantages which are too great to ignore. (Compare Google Maps to the Road Atlas CD-ROM.)
You make a good point about "what happens when we're offline". We suddenly can't use those awesome cloud apps, but that's the nature of the game -- these apps do things which are just impossible for a standalone app to do. Networking is the price of admission.
I hope we'll see a popularization (commercialization?) of mesh networking in the not-too-distant future. Pervasive mesh networking becomes more and more feasible as we have more and more wireless devices around us, and while it comes with its own set of unique technical constraints, it would go a long way to achieve a self-healing network at its outer edges.