I think perhaps, you are assuming too much, is this the only model you can conceive of?
Your stubbornness in insisting that the bishops of Rome were masters of all they surveyed is just making you look silly.
Arbitrators are supposed to be neutral and respected, which is why they are picked. That was why the bishop of Rome was sometimes picked, no See was more respected and trusted.
“Settling matters” is not an evidence of ‘authority over’, especially when looking at how the early church was organized. An awful lot of issues were settled throughout the church by people other than the bishop of Rome, and to this day that process continues.
It has already been explained how the process works. Perhaps you do not have the ability to imagine more than one functional model of organization.
For an example, when the Camp David accords were hosted and facilited by the USA , there was nothing implied, then or now, that President Jimmy Carter had ‘auctoritas’ over Egypt and Israel. But the USA in general and the President of the USA in particular did hold a great deal of respect in the eyes of the two beligerant parties, and that helped pave the way to some otherwise difficult to obtain agreements.
In some parts of Illinois, and I suppose elsewhere, there is a ‘village’ form of government. It is not very common anymore, but the President of the village board is not a mayor, and he does not control the town and has no veto. He chairs the meetings, and gets just one vote like everyone else on the board.
The US Supreme Court is very similar to this, the Chief Justice has one vote like everyone else with no veto. It has always been this way.
When the G 8 meet in their well publicized summits, there is nothing implied that the host, nor the presidency (2009 Italy) of the G 8 actually controls the other countries. The chair is there for structure, but not for control or even direction, the direction and agenda is the concern of every member. They are in all respects equals.
This is how the world works much of the time.