body - Why is there a mechanical shutter in a digital camera?

I understand that in a camera the diameter of shutter opening — the aperture — controls how much light comes in, and this affects the resulting exposure.

But I don't understand why, in a digital camera, the shutter needs to close and open when taking a picture or making continuous shots. Can't the limitations on framerate (frames per second) or fastest shutter speed (for example,¹/₃₂₀₀) just be a property of the electronic sensor?

I ask this because my new new camera can't make more than one shot per second in continuous mode. 1 FPS is ridiculous in a 2011 camera, don't you think? (It can do 30fps for HD video.)