How long it’s “turned on” for is shutter speed. The sensor still needs to be exposed to light for a certain amount of time determined by the user.

Because:

• some still have shutters, especially pro level cameras
• nearly all information about photography involves shutter speed, including software and book tables for calculating exposure, in addition to how cameras communicate with external flash.
• pertaining to the previous point, there are still flashes that are set by shutter speed.
• there are still a lot of film cameras use out there

But really, the simple answer is that shutter speed is just what we call exposure time, for historical reasons. You could call it something else, but it’d mean the same thing. Why rename it, when it’s still literally shutter speed in much of prosumer and high-end photography?

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.

The same reason we hang up cell phone calls. The term means something that people are familiar with and the end result is the same, so we keep using the same terminology even if it isn’t literally accurate due to changing technology.

It can indeed (usually) be turned on and off at will, which is more-or-less equivalent to a traditional camera’s shutter speed.

TL;DR: My mirrorless Canon R10 body has both a physical mechanical shutter and an electronic shutter, and you can choose between the two at will.

There are some additional considerations, one of which is tradition in that early gen digital cameras worked in much the same way, mechanically, as their film counterparts did simply due to ideological inertia. And for digital SLRs (true SLRs, not modern mirrorless cameras) a shutter is necessary due to the physical layout of how the camera itself takes pictures. When the prism is flipped from the viewfinder to the exposure position, the shutter needs to be closed at least for that brief moment so you don’t get a smear of junk reflection imagery swept across the sensor, which would then become part of the picture.

Also, not all cameras, especially early and/or cheap ones, can capture a whole image from their sensor all in one go, so attempting to implement a purely electronic “shutter” may introduce motion artifacts on pictures of moving subjects.

All digital cameras use an electronic shutter scheme when recording video, also.

If you’re asking why they need variable speeds, it’s for the same reason as a film camera. Only so much light can be gathered by the sensor over a certain amount of time, and longer exposures are required to gather enough photons to form an image in low light conditions, and vise-versa. The sensor doesn’t “know” what it’s capturing, it simply dutifully reports – with some degree of inherent error and variability – what amount of light hits each of its cells.

Shutter speed is one of the important choices in taking a photograph for artistic reasons.

It exists in digital cameras because photographers want to be able to make those choices.