Just a random thought that I had... Vader prompted Luke to use his anger in their duel by taunting Luke about Leia. Luke's brotherly love for Leia made him lose control and was ultimately the reason that Vader lost that duel in front of Sidious.

Had Luke been an only child to Anakin and Padme AND Leia been more in love with Han than Luke, how could Vader have taunted Luke into using his anger? How would the duel had turned out? Would Luke have survived against Vader?

Before the duel started, Sidious poked at Luke's anger several times by taunting him about his friends dying and losing the battle. Luke got angry to the point of attempting to strike down the emperor, but he was generally able to control himself. For this reason, I personally don't think that Luke would've survived on the Death Star. If he was pushed hard enough to use his anger against Vader and defeat him, I feel he would've let his anger fuel him and that he would've joined the Sith, whether it be as the Emperor's or Vader's apprentice. Either way, ultimately, Luke dies or joins the Sith in my opinion. Therefore, the empire would win.