Other branches of Christianity absolutely have longstanding centralized governing bodies! The churches of Orthodox Christianity have patriarchates, for example. Protestant and reformist churches, necessarily, have more recent administrations but these can also be quite longstanding, The archbishop of Canterbury, for instance. That said, there are substantial differences in the role of church leadership, especially between catholic and protestant churches--but I'll leave that to experts to historicize.

*Edited so as to better reflect the situation of various Orthodox churches.