Cross-pollination and then seed collection is the traditional method for developing new genetic varieties of potatoes.
Don't bother to save the fruits (and seeds) if that's not your intention. You could pick them off and dispose of them if that floats your boat. Otherwise, they don't appear to divest the developing tubers of energy and materials.
If you really like the variety that you are growing, chopping up tubers to clone more of the same is the
traditional cultivation method. The ingnorantsia call those pieces "seed potatoes" but it would be far more useful to call them exactly what they are = "clone potatoes."