The Anglican church sees itself as standing in continuity with the Catholics although there was a parting of the ways in the 16th century. Your post prompted two central questions for me.
It was a rather blood thirsty parting of the ways with Henry VIII persecuting and killing Catholics, Mary the opposite and Elizabeth and later rulers again persecuting Catholics, not something we hear about much. But be that as it may the Anglican Church does have a lot in common with the Catholic Church, more so that most other Protestant Churches.
I personally believe that they will rejoin at some stage, although there'd be be a lot of issues to be resolved before they do.
1) It was mainly accepted from the time of Peter --> Clement that the Bishop of Rome in consultation with the other bishops could pronounce on doctrine, declare certain positions heretical and clarify the teaching of the church. There is almost universal agreement on the first 4 Ecumenical councils in which such pronouncements were given but not so much since then. If the test of authority is whether something is indeed bound or loosed as a result of an announcement and taking subsequent growth as a sign of Gods blessing or not then it seems certain churches have lost weight over time e.g. Eastern Orthodox,Coptic. Others have prospered for a season and then withered eg many Protestant denominations destroyed by Western liberalism. Others continue to prosper in some places and struggle in others. To what extent does this binding and loosening authority connect with the blessing of churches over the centuries?
The Catholic Church is still going strong. It might appear to be weakening in the West, but it's gaining a lot of adherents in Africa and Asia. which may well become major centres of gravity in the future for the church.
It started with Christ's declaration to Peter and as Christ said, "The proud gates of hell will not prevail against it."
The differences of opinion in the church councils (and I'm not an expert) mainly seemed to be a result of Catholic / Byzantine competition. For a while it seemed the Byzantines had God's approval as Rome became a back water for a season. But eventually things turned around, and Constantinople was invaded and vanquished by the Turks.
Mind you I could see Western Europe becoming Moslem the way things are going, which would leave Rome in a pickle. However the promise was to Peter - not Rome. If the Chair of Peter has to move somewhere else, then so be it.
I have my personal idiosyncrasies and I think it might move to South America at some stage. But that's speculation.
But "to what extent does this binding and loosening authority connect with the blessing of churches over the centuries have to do with the rise and fall of denominations?" I don't know. One of the problems for Protestantism is greater and greater division. I used to be Protestant and I remember my old Presbyterian pastor saying "When it comes to theology, Protestants couldn't agree how far to spit!" And he was Protestant.
As they become more and more divided, the world is taking less and less notice of them. The fact that the media is very quick to criticise the Catholic Church intrinsically shows the world knows which church to take seriously. Protestant churches don't seem to generate as much hostility - not in my opinion anyway.
So as they move themselves further and further away from the "binding and loosing" authority granted to the Holy See, they're becoming of less and less concern to the world.
2) do you accept other dimensions to this binding and loosing in for example in spiritual warfare and church discipline on more ethical than doctrinal matters? Obviously Peters pronouncement to Ananias and Saphira was an example of that. Do you think popes can similarly echo Gods curses and blessings today when rebuking abusive world leaders who claim to be Christian for example?
The problem the Pope would have in rebuking an abusive world leader eg. Joseph Stalin is that the person he would be rebuking would have had a vast military machine and possibly nuclear weapons to boot. This was the problem faced by Pope Pius XII when he was surrounded by Nazi and Fascist forces. I think Hitler was an attempt by the devil to control the world, but he failed that time. So on one hand you had a form of very well armed anti-Christ, complete with Gestapo, SS, the Waffen SS, and the formidable Wehrmacht, and on the other side an unarmed Pope guarded by a bunch of Swiss Guards.
Pope Pius XII and the Holocaust - Wikipedia.
So how much ability did he have to directly interfere in Nazi Germany's actions? Not much.
We therefore might question how much weight Christ's statement to Peter carried. But we forget that the binding and loosing will be bound and loosed
in heaven, not necessarily on earth. We'll find out just how binding that authority is on the day we die, and front up to the judgement seat. Hitler and Stalin might have ruled over millions on earth. But I don't think I'd like to be in their shoes now.
When he was still a Cardinal for example, "Pacelli (later Pius XII) publicly restated the words of Pius XI on the incompatibility of Christianity and antisemitism: "It is impossible for a Christian to take part in anti-Semitism. Anti-Semitism is inadmissible; spiritually we are all Semites" (from the above article).
While this was not a binding statement, his efforts and outspoken protests about the Holocaust would have been noted in heaven. Woe betide anyone who had been involved in the Holocaust on the day of their judgement.
On abortion for example, according to the Catechism -
Formal cooperation in an abortion constitutes a grave offense. The Church attaches the canonical penalty of excommunication to this crime against human life. "A person who procures a completed abortion incurs excommunication latae sententiae,"77 "by the very commission of the offense,"78 and subject to the conditions provided by Canon Law.79 The Church does not thereby intend to restrict the scope of mercy. Rather, she makes clear the gravity of the crime committed, the irreparable harm done to the innocent who is put to death, as well as to the parents and the whole of society.
On the other hand, I was informed by a parish priest that he had the authority to forgive a penitent the sin of abortion. So you might say the authority to bind and forgive sins can be delegated, again via the apostolic succession from pope to bishop to priest viz. "If you forgive the sins of any, they are forgiven. If you retain the sins of any, they are retained." I'd assume that excommunication also means their sins are retained, although I'm not sure of that.
Primarily the authority has to do with the church itself, as I understand it. God was giving His church authority.
Or as the same old pastor I quoted above said to me once as we talked about this very same topic, "What's the use of having a church if you're not going to give it any authority?"