While I am in the first Papal church in the world, I will be the first to admit that nobody – not Orthodox, not Catholics –
needs a Pope. Looking at the history of how this term began to be applied to Bishops, we see that the 13th successor to the throne of St. Mark in Alexandria, HH St. Heraclas, was first referred to by this title in a letter from the Roman Bishop Dionysius to Philemon, which reads in part:
τοῦτον ἐγὼ τὸν κανόνα καὶ τὸν τύπον παρὰ τοῦ μακαρίου
πάπα ἡμῶν Ἡρακλᾶ παρέλαβον.[1]
Pope, Heraclas.]
Leaving aside even the question of what it means to have the Bishop of Rome receiving some rule from the Bishop of Alexandria (I don’t really want to open
that can of worms), it can be plainly said that to go back to a time “before Popes” is at least possible in the Alexandrian reckoning of history, as that would involve looking to how things were before the middle of the third century AD (Dionysius was actually writing after HH St. Heraclas’ death in 246). The Church at Alexandria was, of course, fully functional before this time, and having its bishop be called by this term of affection did not really add anything new in terms of powers to his office. While of course there is deference paid to him as the most senior bishop of the Church and he carries certain responsibilities and privileges in that role (e.g., heading the Holy Synod), I believe that Alexandria does show a good example of a kind of “middle way” between the poles as they are sometimes painted by RC and EO polemicists alike (note: I am not saying that anyone in this thread qualifies to be called a ‘polemicist’, only that the two poles of “the Pope has all kinds of unique powers because he’s the Pope” and “no, he doesn’t; he’s just another bishop” don’t really make sense to me).
As for who is the successor of who, that’s a bit easier in many (well, some…) cases. As some were cast out for various offenses or never accepted in the first place (e.g., the various Arians who were placed in Alexandria by Christ-hating emperors, which are not recognized by either non-Chalcedonians or Chalcedonians in Egypt), the idea that the valid/true/whatever bishop is the one that holds the true faith is not just an easy way to be vague at Catholics in discussions like this
p), but does carry some historical and contemporary weight in so far as both OO and EO have recently and historically had to deal with leaders who were in one way or another not fit for the job (read: since our Popes are not infallible, and we do not share RC ecclesiology anyway, this is actually still a really important criterion by which to judge the health of the leadership, since we cannot assume that anyone is
a priori protected from messing up). It can be fairly said, I think, that this does not provide the safety and security in terms of epistemological certainty that the RCC sees in its system, but from where I’m sitting it’s all the more realistic for that. Bishops do go bad, we know.
So while it is probably fair to say that both sides have their strengths and weaknesses (as conceived by those not in communion, anyway), what has drawn me to Orthodox ecclesiology specifically as practiced in my communion is in the various ancient and modern examples of it in practice. We do not have to hypothesize that such a thing is what was meant in a canon or in some other ancient writing, because the canons are still ‘alive’ and working in the our church, even (or maybe especially) in times of trouble and conflict over just how the Church should move forward at a particular juncture or on a particular issue (e.g., the fight over the application of Canon 15 during the recent Papal election). There’s no doubt it is terribly messy business, and it definitely would be easier if there were one bishop we could turn to to lay down the unquestionable and inerrant law, but what can I say…we may get down in the dirt and look somewhat backwards or not respectable to the more highly organized churches, but there’s something to be said for getting your hands dirty in the process of keeping the narrow path as unobstructed as possible.
When I went away to university, I met a Catholic-turned-Evangelical who challenged me about… well, you know the routine: salvation by grace through faith and works, papal infallibility, the role of Mary and the saints.
So, keep saying your rosaries (I ask that you include me) and everything is peachy. 
(Please make note of the, er, emot-icons.)