"Only a man exceedingly proud and vain," Dumbledore said quietly, as he turned back to the Floo roaring up again with green flames, "would believe that his heir should be like himself, rather than like who he wished that he could be."

That leaves out a third, and more sensible, alternative, which is choosing an heir which is the best person available. That person isn't likely to be an exact match to either the person who's doing the choosing or what that person thinks of as an improved version of themself.