Fundamentally, together with councils, he resolves disputes, especially the ones that threaten to divide the Church. Since even a council can be divided and split, there has to be a buck-stops-here office or else abysmal fragmentation.
The abysmal fragmentation is what happened to those Christians who abandoned a clear authority structure and responsibility. Maybe Jesus didn’t really mean it when he insisted on unity (John ch. 17) but if you want unity, you have to have a way to resolve good faith disputes over how to interpret Scripture. If you don’t like that approach, each preacher gets to interpret Scripture for himself and we end up with 10,000 different “churches.”
Sorta like marriage: either you find a way to resolve disputes or you split up.
‘cuz the one thing you can count on is that disputes will arise. And every single one of the disputants will insist that his position is The Biblical One. So “the Bible as the authority” won’t cut it. Those who took that line now are divided into 10,000 different denominations (some say 30,000 but 10,000 is more accurate).
Or like the whole Anglican/Catholic dispute where you just agree to live in different rooms and both of you drink a lot. :)