I think it is more accurate to say that the sensor is always on, there is an inherent activation/purge speed in any semiconductor (though it is fast, especially with modern tech compared to early digital cameras), but it is always on, we just aren’t always paying attention to what it is saying. There is an issue that it takes time to convert the information from the sensors into something useful to the device. You can store the time series output of a light sensor but this isn’t particularly useful to photography, still or motion, and if you want to transmit it you would need one line for each pixel. Instead we encode it which relies on sampling the sensors.

There are also considerations on how you want an image to look. Photographers will adjust things like exposure length and sensitivity to get the shot they want. Longer exposures collect more light, but you get more motion-blur. Shorter exposures the opposite. if you want to say, capture an image of a fly, you would go fast so you can get crisp detail before it moves. If you want a star you will go slow and use tracking to prevent too much blur but get loads of light. If you are a film director and want to give an impression of the speed of something, you might want to shoot longer to emphasise the blur.