Why is it that neither the Sunni nor the Shia have elected a new Caliph? Wouldn't the Sunni hold an election, or the Shia choose a descendent of Muhammad as the new Caliph? Is there any reason that the Muslim world has decided against having one since 1924?
"Legitimate" according to whom? I guess one of the main problem is, it is terribly difficult to get all the countries in the Muslim world, along with their citizens and leaders to just abandon their sovereignity, and to find a leader and a set of laws that they can all agree with.
There were movements to revive the Caliphate after 1924. The most prominent being a marriage between the daughter of the last caliph and the son of the richest man in the world (the Nizam of Hyderabad). The idea was to merge the money of Hyderabad with the political clout of the Ottomans. The plan was a failure though; both the Nizam's son and the caliph's daughter were far more interested in being socialites than ascending to political supremacy. A few other movements took place but none really bore fruit.
Is there any reason that the Muslim world has decided against having one since 1924?
I would disagree that the Muslim world decided against it. After many unsuccessful attempts, the movements sort of petered out and Muslims saw it as a more long term goal. In the meantime, there simply hasn't arisen any one with a legitimate claim or one that Muslims would support.