St. Peter’s account of being labelled head of the faith is mostly just Catholic doctrine. Whichever of the apostles ended up in Constantinople, then Byzantion, Jerusalem, Bethlehem, and Alexandria could theoretically have claimed similar visions and been taken just as seriously because before power was consolidated Christianity was “run” by a Pentarchy of Patriarchs, one in each of the five holy cities of the faith, and each of whom technically equalled the pope in rome in rank, just in the sense that pope’s descend from St. Peter while other patriarchs descended from different apostles or early converts.
Had Christianity spread in India you’d probably hear about a Christian Hexarchy with one of the patriarchates based in Chennai.