Depth doesn't come from complex controls, but complex controls are almost always needed to be able to play a game with any depth.
With mobile touch-screen controls the game has to somehow register a button-press as soon as a button-press is made.
But it must also never (to a first approximation) register a button-press when a button-press was not requested (e.g. an accidental brush of a finger against the screen).
It's hard enough to do this anyways, but the "input device" does no favors for the game developer either. There's a reason that input devices on consoles and computers have evolved over the years into the shapes they have, they are meant to be easy to hold in a certain position (so that you always know where the buttons are) and perhaps more important, they have tactile features that allow your finger to quickly find and stay near the buttons. It's the reason that buttons and the D-pad are not fully-flush with the shell, even though that might have been more stylish.
Imagine trying to play Super Mario World with a NES-shaped rectangular controller where there are no tactile buttons. This might have worked if you only needed Left, Right, Jump, but even that game required all of the following for the best gameplay: Left/Right, Up/Down (for pipes and to literally scroll the screen up), Jump, Run/Hold/Fireball, Spin Jump, Pause, Drop Stored Item and Shoulder Left/Right (to scroll the screen). And if the input wasn't right you'd either jump into a spike you had meant to run under or accidentally run off a cliff you'd meant to jump past.
The controls on Mario seem complex only because they were so completely intuitive when you were actually playing them but I assure you they were far more difficult to implement than you think. It's not until you start playing platformer games with horrible controls that you can realize the great job done by the development teams at Sega (for Sonic) and Nintendo.
The concept of an on-screen virtual analog stick is great in theory. I practice, its always going to be terrible in comparison to its physical counterpart. Games like Modern Combat 4 are pretty impressive when you first see them, but 5 minutes in you realize two things: The controls are bad, and the entire game has been dumbed-down because the developers know that the controls are bad. Its not even their fault. They did as good of a job as anyone has, but a capacitive touchscreen just isn't going to offer the user a pleasant experience in a 3d fps.
Certain genres are well-suited to touch controls. Many genres are not. Board games and card games work very well.
I'd agree that board-game style strategy/war games would work, and also quite a few turn-based strategy games.
Its also possible to make certain types of real-time strategy games. Unfortunately, complex RTS games like Starcraft probably can't be done well without extreme oversimplification.