Let me add some images.

On a mechanical shutter, you have two physical curtains (A) that move when you take a photo (B). The shutter speed is the average time the gap is over a zone of the sensor (C). In this example is half (Yellow) of the sync speed (Green)

When an object is moving fast, (and you have sufficient light to capture the image) the shutter gap is narrower to leave less light passing through it (D), but the curtains move at the same speed. If the object is too fast (E) it will keep moving relative to the gap, leaving a rolling shutter effect (F).

The maximum sync speed is when neither of the two curtains is over the sensor, and it is fully open. You use this when using a flash. Opening the first curtain (G) and when they are fully open you trigger a flash to capture flash (H) and then the second curtain starts closing (I).

If you use a faster shutter speed than the sync speed with a flash, (O) this will only illuminate the open zone (P) with probably some exposure on the rest depending on the ambient light.

In the image (C), if your maximum sync speed is let us say, 1/250, as it is half the opening, it would be 1/500. If the gap is 1/4 it would be 1/1000 and so on.

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Some electronic shutters have a similar rolling effect because of how the sensels of the sensor are processed or read (J).

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Do fast shutter speeds actually work?

If your camera is working properly, yes, it would work if you have sufficient ambient or continuous light like sunlight. It would expose each zone of the image at 1/4000 of a second.

It will freeze the movement to some extent depending on the factors I mentioned.

Remember that a fast shutter speed is not only used to "freeze movement" but also to properly expose images, for example with large apertures for bokeh effects.

Additionally, there are HSS flashes, that basically make a "stream of flashes" so the duration of the stream is enough to cover the time the gap covers the sensor. Those can be used in combination of a fast shutter speed.

You need a large aperture for bokeh so you need a fast shutter speed (K). You need a flash to properly expose the subject (L) You use an HSS flash (M).

To freeze some other type of elements you need a studio setup.