This is not true of games. Can you name any? Yes, multiplayer games require a server component, but they're not thin clients. And games that are not multiplayer generally don't require a server at all except possibly for DRM. People don't like wasting their data plan unnecessarily and they like being able to play in subway tunnels.
If you think about the economics of it, this makes sense. Margins are razor slim; you need to be able to profit off of whatever is left after the app store takes its cut of your $0.99-4.99 retail price. Having to run and maintain a fleet of servers to handle your games simply doesn't make sense. Better (and easier programmatically) to take advantage of the powerful processors on the devices themselves and have the games run locally. Think of something like Civilization, that has a mobile app; why on Earth would you want a central server that would have to run god knows how many millions of AIs when the phones could simply calculate that themselves?