I’m not sure what you mean by “lose their flavor”. I had pomegranates in January from abandoned trees and they tasted great. Obviously, if a pomegranate splits, it becomes susceptible to elements and insects that are likely to cause spoilage and rot, so I do not recommend leaving split poms on a tree. However, unlike figs that go from ripe to overripe in a day, I do not believe that there is such a thing as an overripe pomegranate.
I should add that sometimes, especially on young pomegranate trees, there are small fruits that develop late and never fully ripen, they would just dry out if left on the tree. This special case is an exception from the statement above.