I think there are two main reasons:
Don't underestimate inertia, especially when you're talking about establishing a position with such a worldwide implication as the Caliph. For example, it's relatively a simple matter for a 19th-century Ottoman Sultan to say that he is the Caliph, given that for the previous hundreds of years all Ottoman Sultans are caliphs. Or, for As-Saffah, he could argued that he had conquered the Umayyads, who were previously the caliphs, so he should be caliph as well. Almost all universally accepted caliphs could claim some kind of continuity that made them the successor of the previous ones.
In contrast, there hasn't really be a caliph since the abolition of the Ottoman caliphate. Someone wishing to claim to be caliph today will have much harder challenge to convince everyone else that he is the caliph.
Since mid (or early?) 20th century, Muslims have become nation-states, with their own independence, government, head of state, constitution, etc. This wasn't the case during the old caliphates. A caliph is not only a spiritual leader but also the supreme political leader of the entire Muslim community. How would it work with nation states? Imagine you have a candidate for caliph, can you imagine that Muslim nations, such Morocco, Turkey, Saudi Arabia, Pakistan, Indonesia would all be willing to amend their constitutions, cede their sovereignty, to declare allegiance to this caliph? Who would be eligible to be Caliph and how would he be elected? These questions are hard to answer today.
As an additional note, since many readers here likely have Western or European background, let me ask you this question: Why hasn't been any Holy Roman Emperor since 1806? I think the answer would likely be similar.