There exists a weird notion out there that (tactile) apostolic succession is part of the normative experience of Anglicanism. I must say that apostolic succession is a vital part of the historic faith of Anglicanism, but it is not typically understood in the fashion that many of these neo-evangelicals understand it today. I added in parentheses in the previous sentence the word "tactile" which is key to understand the complaint I am expressing. The "tactile succession" refers to the practice of having previously consecrated bishops consecrate new bishops, typically three bishops in number will consecrate a new bishop. This is often associated with Roman Catholic and Eastern Orthodox doctrine because it is in this succession that they believe bishops to derive their authority and sacerdotal powers. However, the tactile succession does not have to be tied to Romanist or Byzantine doctrine, as it is practiced in Anglicanism innocently without such association. However, it is not a necessary part of what it means to be an Anglican (for those doubters, one should consult with Archbishop Laud who deemed Lutheran [appointed] superintendents as valid bishops as well as royally "consecrated" [=appointed] Scottish bishops as valid). To distinguish non-Romanist tactile succession from a sacerdotalist understanding, I tend to employ the term historical succession, rather than "apostolic" for the former term conveys that which we wish to express in this act. The historical succession is a visible reminder that we are passing down the apostolic faith through the teaching and preaching of the bishop (the real apostolic succession). It gives us a sense of heritage and continuity to see bishops consecrated by their elders in the faith. In this sense, the historical succession is important to our experience as Anglicans. Yet, it is not equally emphasized in Anglican church parties, it was emphasized more by the Caroline Divines than the typical Church of England man at the time. As I had mentioned earlier, even the Caroline Divines did not regard the historical succession as integral to the faith, considering it was not absolutely necessary.
The Reformers noted many disparities between the (then) contemporary practice of episcopacy and the nature of the apostles in the New Testament. They had many criticisms directed at the pope (which will be mentioned briefly later). First, they noted (as later scholars would) that the polity of the church in the New Testament was fluid and that it was impossible to identify a pattern of church governance which matched completely with the practice of the NT Church. For this reason Anglicans have typically espoused the bene esse position, or that no form of church governance is absolutely necessary for the existence of the church, because of this "problem" in scholarship. If one cannot identify with relative certainty the polity of the NT, it becomes rather difficult to enforce a particular option among many. Consequently, many Reformers attacked the prelacy of the Church at the time. This refers to the practice of monarchical episcopacy or the practice of assigning a bishop a diocese with a see. They noted the itinerant nature of the apostles' ministry in comparison with the princely rule of the contemporary bishops. Of course, this is not to say in condemnation of prelacy, for the Church of England maintained prelacy. However, one daughter Church attempted to disavow it (unsuccessfully, considering modern results), that being the Protestant Episcopal Church in the United States of America. PECUSA did not have dioceses until the 1830s and did not have cathedrals until the 1880s.
To get to the heart of the matter, apostolic succession doesn't refer to the consecration of bishops but the preaching of the Gospel. We can know that we are Catholic because we "profess sound, solid, and pure doctrine" (Whitaker), not because we maintain a neat list of consecrated bishops. As Whitaker explains in other places, "[W]e regard not the external succession of places or persons, but the internal one of faith and doctrine" (Whitaker to Bellarmine). The Reformers were concerned with a maintenance of Catholic doctrine, which had been abandoned by Rome, not the lists of consecrations, which Rome had dutifully kept. It is in this pure preaching of the Gospel that we can avoid false doctrine, as John Jewel stated, "'Succession,' you say, 'is the chief way for any Christian man to avoid antichrist.' I grant you, if you mean the succession of doctrine" (John Jewel to Thomas Harding). Dean Philpot of Winchester states in his answer to a question related to the (perceived) lack of succession in the English Church:
I deny, my lord, that succession of bishops is an infallible point to know the church by; for there may be a succession of bishops known in a place, and yet there be no church, as at Antioch and Jerusalem, and in other places where the apostles abode as well as at Rome. But if you put to the succession of bishops succession of doctrine withal (as St. Augustine doth), I will grant it to be a good proof for the catholic church; but a local succession is nothing available. (Philpot)The guarantee of the church is not rested upon lists of successors, which is principally bound up in faith of men. Rather, the apostolic succession which we profess is the holy and pure Gospel of Christ, that catholic faith which we profess in the Athanasian Creed, that faith which saves us from eternal damnation and separation from Christ. This is the apostolic succession which is crucial for the existence of the church and the foundation of Anglicanism.
I mentioned the pope earlier, while I did not intend this to revolve around the papacy, the Reformers did have criticisms of the papacy, especially in relation to its assertion that it was the successor of Peter. Here follows an excerpt from Jewel's Apology, treating on the pretended nature of the papacy as the successor of the Apostle Peter:
"[W]hereas some use to make so great a vaunt that the pope only is Peter's successor, as though thereby he carried the Holy Ghost in his bosom and cannot err; this is but a matter of nothing and a very trifling tale. God's grace is promised to a good mind and to one that feareth God, not unto sees and successions
"Yet notwithstanding, because we will grant somewhat to succession, tell us: Hath the pope alone succeeded Peter? and wherein, I pray you? In what religion? in what office? in what piece of his life hath he succeeded him? What one thing (tell me) had Peter ever like unto the pope, or the pope like unto Peter?
"Except peradventure they will say thus: That Peter when he was at Rome never taught the gospel, never fed the flock, took away the keys of the kingdom of heaven, hid the treasures of his Lord, sat him down only in his castle in St. John Lateran, and pointed out with his finger all the places of purgatory and kinds of punishments, committing some poor souls to be tormented and some others again suddenly releasing thence at his own pleasure, taking money for so doing; or that he gave order to say private masses in every corner; or that he mumbled up the holy service with a low voice and in an unknown language; or that he hanged up the sacrament in every temple and on every altar and carried the same about before him, whitersoever he went, upon an ambling jennet, with lights and bells; or chalices, churches, and altars; or that he sold jubilees, graces, liberties, advowsons, preventions, first fruits, palls, the wearing of palls, bulls, indulgences, and pardons; or that he called himself by the name of the head of the church, the highest bishop, bishop of bishops, alone most holy; or that by usurping he took upon himself the right and authority over other folks' churches; or that he exempted himself from the power of any civil government; or that he maintained wars and set princes together at variance; or that he, sitting in his chair; with his triple crown full of labels, with sumptuous and Persian-like gorgeousness, with his royal scepter, with his diadem of gold, and glittering with stones, was carried about, not upon a palfrey, but upon the shoulders of noblemen.
"These things, no doubt, Peter did at Rome in times past and left them in charge to his successors, as you would say, from hand to hand; for these things be nowadays done at Rome by the popes, and be so done as though nothing else out to be done.
"Or contrawise, peradventure they had rather say thus: That the pope doth now all the same things which we know Peter did many a day ago; that is, that he runneth up and down unto every country to preach the gospel, not only openly abroad but also privately from house to house; that he is diligent and applieth that business in season and out of season, in due time and out of due time, that he doth the part of an evangelist;...that he doth not feed his own self but his flock; that he doth not entangle sovereignty over the Lord's people; that he seeketh not to have other men minister to him, but himself rather to minister unto others; that he taketh all bishops as his fellows and equals.
"...Unless therefore the popes do the like nowadays as Peter did the things aforesaid, there is no cause at all why they should glory so of Peter's name of his succession." (Jewel)