The reason for the pope/papacy is because God established the position, that's all the reason Catholics need...
The Church and the Sovereign Pontificate
exist because God established them, but answering as you did
completely avoids the question of WHY, what is the purpose, the
raison d'etre of said institution/position. Your answer is that of a child, by which I mean no offense to intelligent children.
God established them, yes. WHY did He establish them?