Briefly, how did the Catholic Church end up with a single head of heads (the pope)?
Because that's the way Jesus set it up.
The Church is founded on Peter, that is explicitly stated in Matthew 16:18. Then we have further supporting references in Luke, and John.
In Acts, it's is clear that Peter is the head of the Church — even Paul, who nevertheless argued with Peter when he thought Peter was giving out the wrong signals (Jewish Christians eating apart from Gentile Christians).
We also have material evidence that baptism by a disciple (including those 72 sent out before Christ's arrest) was not 'full', and that the full rite was tripartite (Father, Son and Holy Spirit) and was performed by an Apostle — ideally Peter himself.
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The shift of the 'Centre' from Jerusalem to Rome is signified a number of ways, from the rending of the veil of the temple, to Paul's mission, and the greater mission of the Church.
Rome, as an 'apostolic see', was always accorded priority over the other sees, (Jerusalem, the fourth, having been destroyed in 70AD) and letters from the time, such as that of Clement of Rome to the church at Corinth, assumes a certain authority in spiritual direction.
Likewise the other churches appealed to Rome for a ruling when they could not resolve disputes internally.
In the Councils, the representatives from Rome were always given pride of place, and spoke first.
The practice was universal until schisms set in.
So he papacy and the priesthood is not an invention, but rather their absence in post-Reformation denominations (Some Oriental patriarchates refer to their head as 'pope, the Coptic Church, for example), speaks of later decisions to do other than Our Lord intended, just one example of how the assumption of Christianity might well be incomplete, if not entirely assumed.
Thomas
