On the screen at the back of the camera, a preview is constantly shown. I assume this preview is made by light hitting the sensor constantly. So this must require the shutter/aperture to be permanently open.

Correct. This puts the sensor into what's called "Live view" mode which is effectively the mode it takes video in. Data is constantly being read from the sensor over and over again.

So if light is constantly hitting the sensor, what's the point of a shutter?

In live view mode most sensors cannot achieve shutter speeds faster that a certain speed, let's say 1/100s (but it could be as slow as around 1/30s, just depends on the sensor design).

This is because in live view mode, on a CMOS sensor, the data is progressively cleared from the sensor, line by line, from top to bottom. This is a process that takes some time and must completely end before reading from top to bottom again.

A physical shutter ensures that you can use faster shutter speeds, by blackening the sensor while it does its initial wipe/read, exposing for a short time, then blackening the sensor while the data is read from it. The physical shutter itself can achieve speeds faster than 1/200s because it moves fast, but also because it can start to close the second shutter curtain before the first one is fully open. They're typically able to reach 1/1000s or faster (even 1/4000s in some cases) doing this.

That said, there are some sensors that have a feature called "global shutter" whereby the electronics in the shutter can effectively freeze the state of the shutter in time instantaneously allowing for faster effective shutter speeds even without a physical shutter. These can often reach higher speeds than physical shutters (ie faster than 1/4000s). If your camera has this type of sensor, a physical shutter is not needed.