How has apostolic succession been "unbroken" in the wake of depositions, antipopes, immorality, and simony? In studying the history of the papacy one discovers all sorts of events that are interesting when considered in the context of apostolic succession from St. Peter until today, including, but not limited to:


Instances of popes being involuntarily deposed by conquering sovereigns and replaced with one sympathetic to that conqueror.
Instances of contention over the papacy by two or even three popes simultaneously, accompanied by multiple colleges of cardinals and bishoprics.
Instances of nepotism and dynastic possession of the papacy by aristocracy and politicians.
Instances of popes engaging in orgies, conspiracies, accepting bribes, fathering children, and even committing murder.
Instances of the papal office being sold. 


Is the nature of apostolic succession meant to be an unbroken succession of valid popes? How does the Catholic Church address these events and the manner of succession when making the claim (from New Advent):


  The fact is indisputable: the Bishops of Rome took over Peter's Chair and Peter's office of continuing the work of Christ. To be in continuity with the Church founded by Christ affiliation to the See of Peter is necessary, for, as a matter of history, there is no other Church linked to any other Apostle by an unbroken chain of successors.
Apostolic succession and Papal succession are not one and the same.

Is the nature of apostolic succession meant to be an unbroken succession of valid popes?

No. Rather the nature of apostolic succession is meant to be an unbroken succession of valid bishops.

Not just the pope, but all Catholic bishops have unbroken apostolic succession. In other words, for a person to become a bishop (even the bishop of Rome - aka pope), he has to be consecrated (Laying on of hands) by another a validly consecrated bishop. Who in turn should have been validly consecrated by another validly consecrated bishop...... If we keep backtracking like this, we will end up with one of the thirteen apostles (The Twelve + Paul). This is considered to be unbroken line of apostolic succession.

Popes might validly succeed each other if and only if they have proper apostolic succession (i.e they are validly consecrated bishops). This is why Stephen II is not considered a pope.

The quote you have mentioned talks about continuity in the see of peter. That is the continuity in papal succession. In other words, there is no time in history of papacy where the see of peter (or the post of the pope) is deserted with no claimants and no conclave was held to elect his successor. Note that the see could be vacant, but during that time the conclave could be taking place. This is what is called unbroken line of successors of Peter.

Instances of popes being involuntarily deposed

New popes are not considered valid unless the previous pope submits to his authority. (As in deposing of Pope John XII and Pope Leo VIII's invalid first election)

Instances of contention over the papacy by two or even three popes simultaneously

Even though at the time when this was happening, it was unclear as to who is the legitimate successor, we can be sure that at least one of them was legitimate. In other words, the see of peter was not vacant. (Retrospectively we can determine the legitimate successor.)

Instances of nepotism and dynastic possession ... popes engaging in orgies, conspiracies, accepting bribes ....

Sinfulness of the person holding the office does not invalidate the office itself.

Instances of the papal office being sold.

If a reigning pope resigns for money, it is a sin (Simony). Like all sins this is also a voluntary act. Hence he will cease to be pope.