The purpose of the potato is a dormant storage tuber so the plant can hide out the winter underground then start into growth next year. So once the plant has stored all the goodness from the leaves, the tuber stops growing (and it wouldn't be able to produce more "food" as that requires photosynthesis, which needs light, which needs being above ground).

The tuber stays underground, under attack from slugs and other pests. Then in spring, probably triggered by soil temperature, it starts putting out shoots, and converting the stored food into a new potato plant.

So from your point of view there is no point in leaving the potato underground once the top has died back, and very good reasons (slugs) to dig them up.