Basically One Piece is intended for "shonen" audience, namely teens and young adults. The time in life when you start watching One Piece is the time when you start saying goodbye to your child-parent relashionships as they were, and start shaping a new adult-to-adult relationship with your parents.
Adolescense is a process through which you tend to lean more to your friend's advice and find yourself in arguments with your parents, until you finally shape your own adult personality.
In One Piece, you can find the same process with many characters with a "father issue" like Ace, Yamato, Sanji and others. Choosing friends over family is what many of the One Piece watchers are experiencing themselves at that specific time in life, so they can relate to it better.
At a later time in life, when you have chosen and established your own family, family becomes more important again.