Correction: the _integral domains_ for which every nonzero element is a power of a fixed nonzero nonunit element up to associates are exactly discrete valuation rings.
So my final statement should read:
"...which again doesn't make UFDs more relevant than any other condition you might add _that is stronger than the much more general condition of being an integral domain_"
So my final statement should read: "...which again doesn't make UFDs more relevant than any other condition you might add _that is stronger than the much more general condition of being an integral domain_"