But didn't Peter have a unique role? *Some* sort of primacy.
Sure, Peter had "some sort" of primacy among the apostles. We don't reject that at all. What is at issue is whether or not Christ intended that this primacy continue among their successors by having one specific bishop to whom all other bishops look as "the first", as "the leader". You don't find this in Scripture. You have to assign it to the Popes by analogy: since the Bishops of Rome are successors to St Peter in the place where he died, they have this primacy among the bishops of the world. But you don't find that in Scripture either. Neither do you find it in the early Church: when I was interested in becoming a RC (a period of a few years), it was reading church history that convinced me this "primacy", as RC's have defined it, simply did not exist before the fifth century. Even after that, it was much less than what Rome has made it at present.
What exactly has the Roman Church stated that is *against* the faith? Not just what is perceived to be an "innovation" but actually against the faith.
What do you mean, "against the faith"?
Doesn't it just seem logical that Jesus would want the body of believers to be under one figure who shepherds the flock like Him?
Logical? Based on what?
I don't think it's an accident that Rome's claims to primacy increase as it became further and further alienated from the rest of the Church. First of all, Rome was the only "apostolic see" in the West, and so it was natural for all other bishops in the West to look to Rome (unlike in the East, where there are a number of "apostolic sees"). Gradually, as the West and the East drifted apart, both sides viewed the other as the break-away and themselves as "the true Church", the legitimate heir to the one Church. As time goes on, Rome continues to proclaim itself as "the true Church", continues to expand through noble and ignoble methods, and by the nineteenth century is found all over the inhabited world. At that point, you're not dealing with a "local Church" in Western Europe vs a communion of "local Churches" in the East, you're dealing with a global entity vs those "local Churches" in the East. It's roughly in tandem with this development that you have these ideas about "Catholic" meaning "found everywhere" or the logic of having "one figure who shepherds" what has become a global flock. None of these things are biblical, apostolic, patristic, or what have you. They are developments that snowball from the estrangement of the West from the truth.
As an ex post facto explanation for how the RCC is organised, yes, their Petrine claims are quite nice, logical, "fitting". But it flies in the face of Scripture and the totality of what we know from history, from patristic writings, conciliar declarations and legislation, etc.