The evidence certainly favors the views of the Churches who uphold Apostolic Succession. Briefly put, that is the view that the Apostles of Jesus constituted the first bishops of the new Church, from which all other valid bishops are since descended by the laying on of hands; or, if you like, that the first bishops were appointed directly by the apostles in the founding of new churches, to act as their surrogates and successors. In 381, the view of bishops as the successors of the Apostles would have seemed obvious; indeed, what we call "the monarchical episcopate" had been described by Ignatius of Antioch in the early 2nd Century.
Now, there are some serious quibbles to be made here about what exactly a "bishop" was in the New Testament, and what Ignatius thought a "bishop" was, and then what later bishops and theologians came to see them as. For me, I see in this a serious case of "back-formation," as when people of a later time re-interpret the past in terms of the present. So, a 4th Century bishop -- or a 14th Century bishop -- will tend to see himself (including such powers, duties, and relationships as he is used to seeing as part of his office) being described whenever the word "bishop" occurs in a record from times past, even if the powers, duties, and relationships of bishops of those days were rather different.* But even granting this, ancientness of usage will hallow even a (partially) mistaken concept. The idea that Apostles = Proto-bishops = heads of organized dioceses or patriarchates has been around for a very long time, and was adopted very early on.
But what was it that later bishops could claim to have received from the Apostles? If we were merely talking about a historical claim -- that is, our Church is recognizably the same in origin, in doctrine, in worship, well that's one thing. What we're saying then is that if an Apostle walked into our church, he'd recognize us as the fruit of his witness and a continuation of his work. In this sense, Succession = Authenticity.
But what is often argued by proponents of Apostolic Succession is a sacramental view of ordination that makes of the Holy Spirit a quantum to be passed on through the legitimate laying on of hands by the ordaining bishop. Those who are irregularly ordained do not possess this quantum; therefore, they cannot pass it on. It's like running a relay race. If you cross the finish line without the baton, you're disqualified, even if you finished first; therefore, the hand-off is of great importance.
This quantum view of the Holy Spirit I find difficult to hold. The sacramental theology of the Middle Ages (at least in the West) became ever more concrete, and this kind of thinking dominated theological expression; nevertheless, it was never systematically applied. Rulings on accepting baptism or ordination by schismatics and heretics went sometimes one way, sometimes another. Rulings by some authorities on the validity of episcopal consecrations have varied based apparently on the whim of the authority (as in the RCC recognizing the orders of the Old Catholics as being valid and in succession, but not those of the Anglicans).
But doesn't authority get passed along with the laying on of hands? Of course. But authority is one thing, the Holy Spirit another. I think someone could be irregularly ordained, and still receive the Spirit of God; likewise, the fact that one is regularly ordained is not a guarantee that the one having hands laid on him is in fact open to the Spirit. The Spirit is sovereign; he condescends to the rite, yes, but he is not a quantum to be passively transferred. "It seemed good to the Holy Spirit and to us" -- the formula of the Council of Jerusalem -- is still the way the Church works. Human judgment and divine blessing work together, but this is not a mechanical thing.
I'm not here endorsing the Donatist view, that the efficacy of sacraments depends upon the character of the celebrant. The Spirit does condescend to the rite. Nor am I saying that faith must be perfect in the recipient -- a pernicious idea much troubling to seekers in the revivalist Churches that constantly ask "Are you sure?" But I am saying that in the end, I am willing to accept a fair amount of human irregularity in the conduct of the Church's business. And I think God will work with and through that.
I have a tremendous respect for ancient institutions, and think new = better the mantra of a perverse generation. But the history of the Church is nearly twenty centuries long now, and many things have been done in many different ways; nevertheless, the unity of Christian thought, doctrine, piety, worship, Scripture is self-evident. I want to be as generous and inclusive as I can be in defining apostolicity. And I see the touchstone of it a demonstration of connectedness with the past, reaching even to the Apostles.
* I note that as the powers of bishops expanded over the centuries, their ability to act as pastors necessarily shrank. Their colleagues, the presbyters, assumed more and more functions formerly performed by the bishop (alone or with their assistance). The bishop could no longer celebrate the eucharist in all his congregations. He could no longer baptize all the new converts. Only by keeping dioceses small (or adding suffragan bishops) could he even hold on to doing most confirmations. The history of the episcopate is one long negotiation with the demands of the office, with the end in view of holding on to as much prerogative as possible. With this in mind, seeing bishops as the successors of the Apostles is very useful in their not being viewed as superfluous to the life of the individual parish, which may have seen them only occasionally.