If you want to get technical: Water is complicated. There are a whole bunch of things going on. The dominant effect is that the bulk pool water is typically above freezing and has a much slower cooling rate than water sitting still in the pipes. Keeping the water well mixed by running the pump keeps the water in the pipes above freezing despite the higher cooling rate in the pipes compared to the bulk pool water by spreading out that cooling across the bulk pool water.
But there are several additional effects that don't tend to be significant around pools, but do make any "absolute" answers wrong more often than not. The simplest effect is that it takes time for water to freeze. While that is happening, the frozen water typically stops moving. So it isn't so much that the moving water doesn't freeze, instead it is that the not yet frozen water keeps moving.
There is also heating caused by the movement/friction of the water. This effect is very small, but can be enough to keep moving water from freezing despite the environment being below freezing, at least as long as the temperature differential isn't all that high.
It is also possible for the required nucleation sites to get preferentially removed from the moving water, allowing the moving water to become super cooled. This effect often occurs in fast moving mountain streams, allowing them to be below 32 degrees and not freeze (i.e. super cooled). In these situations, freezing still progresses from the outside in, so that the stream will eventually freeze solid, but this can take some time, potentially several weeks if the conditions are just right. One effect that enhances this one is that ice is a good insulator, so the outer layer of ice slows the cooling rate of the moving core by reducing heat transfer to the environment.
There will also be water that never freezes due to differential concentration of impurities. As the bulk water freezes, impurities become concentrated in the remaining liquid water, eventually accumulating enough to lower the freezing point below the ambient temperature in nearly any plausible situation. This has dramatic effects on polar ice sheets, as life becomes concentrated in the remaining liquid water, which tends to form channels that allow far more movement and ongoing development of that life than most would expect in polar conditions.
But none of that really matters in swimming pools, where all you need to know is that running the pump does indeed prevent freezing of the water in the pipes for a time, but not forever. There is a whole band of moderate climates where it does freeze, but the amount of time you gain by running the pump is long enough that spring comes before there is any problem. At least as long as the power doesn't fail.
