Part of the confusion comes from the fact that consigliere translates into counselor or advisor and people assumed that it means "to the Boss". The way I understand it,because of numerous abuses by Bosses from the Mustache Pete days,the Organization thought it would be wise to have a person who would serve as sort of a court of appeal for the rank and file guys.

Since the Underboss isn't going to make waves,and no Capo would have the authority,(or the stones) to challenge a Boss's decision to whack one of his guys on a whim,the consigliere is supposed to serve as a check and balance against that sort of thing.

Like a lot of Mob policies,it's not 100%,but generally,the spot goes to someone who is seen as level-headed,and loyal to the Family,not necessarily the individual Boss.