Peace be with you. Short answer at the bottom.
The doctrine of ancestral sin and pelagianism only appear superficially the same as you have presented them. (especially considering the wikipedia article is very generalized)
Concerning Ancestral Sin
Here is a real life example of how ancestral sin does not make you guilty be default.
If you were born from the family of Hitler, you would not be guilty of genocide. You would only be "associated" with genocide due to the actions of someone in your bloodline.
Similarly all of humanity is "associated" with sin due to the actions of Adam and Eve.
So just like "Alexander Hitler" is forever tainted by the actions of Adolf, we are forever tainted by the actions of Adam & Eve. Alexander cannot really live without the consequences of someone else's actions looming over his life. This is true for us as well, the consequences of sin was "death" and "separation from God" and these consequences loom over our entire life.
Now Alexander can attempt to be a good person, but he cannot ever restore his family name to one that is not reviled. And the same is true for us, we cannot correct the stain on human nature that Adam gave us.
Fortunately we have God to help us. Jesus with the dual nature of being fully Human and fully Divine has corrected this stain. He has allowed us to have a personal relationship with God again. (We can call God "Father", before this was not possible)
Without going into detail, there are other questions that address this, God has corrected the consequences of sin through His only begotten Son.
In conclusion... Alexander or any of us are faced with choices where we can attempt to be virtuous or not. Alexander could "change his last name" or he could live with it and be a virtuous man. The same is true for us... we can "run away [from our problems, from God]" or "take on responsibility and seek God"
Concerning "Isn't it possible to simply not sin?"
The answer is yes. The Theotokos is accepted to be sinless.
Is it hard to believe that the Theotokos did not sin? If we consider this from a merely human, logical, and rational point of view it is indeed difficult. On the other hand if we consider this question in the context of the history of salvation, and eternal life in God’s kingdom, it will be easy to accept. The incarnation of Christ and the deification of man was God’s plan of salvation from all eternity, “the mystery which hath been hid from the ages” (Col. 1:26). Who and what did God the Father want the woman who would give birth to His Son to be? Could it be other than the highest of all creation, ever-virgin, and sinless. This is so logical, this has to be. -Monastery of St. Tikhon of Zadonsk
While we do believe the Theotokos had no actual sin, she was born, as were all of us the descendants of Adam, with the effect of sin on our human nature.
In the writings of St. Silouan the Athonite we read: "In church I was listening to a reading from the prophet Isaiah, and at the words, 'Wash you, make you clean,' I reflected, 'Maybe the Mother of God sinned at one time or another, if only in thought.' And, marvelous to relate, in unison with my prayer a voice sounded in my heart, saying clearly, 'The Mother of God never sinned even in thought.' Thus did the Holy Spirit bear witness in my heart to her purity." -Archimandrite Sophrony (Sakharov), Saint Silouan the Athonite, p. 392
But wait how? How to not even sin in thought?! It seems impossible, but there is more detail to be had regarding this matter.
"Mary was held in admiration by all the people of Israel; and when she was three years old, she walked with a step so mature, she spoke so perfectly, and spent her time so assiduously in the praises of God that all were astonished at her and wondered...She was so constant in prayer, and her appearance was so beautiful and glorious, that scarcely anyone could look into her face...And this was the order that she had set for herself: From the morning to the third hour she remained in prayer; from the third to the ninth she was occupied with weaving; and from the ninth she again applied herself to prayer. She did not retire from praying until there appeared to her an angel of the Lord from whose hand she used to receive food; and thus she became more and more perfect in the work of God. Then, when the older virgins rested from the praises of God, she did not rest at all; so that in the praises and vigils of God none were found before her, no one more learned in the wisdom of the law of God, more lowly in humility, more elegant in singing, more perfect in all virtue. She was indeed steadfast, immovable, unchangeable, and daily advancing to perfection...She was always engaged in prayer and in searching the law...." -The Gospel of Pseudo-Matthew, The Ante-Nicene Fathers, The Writings of the Fathers down to 325 A. D. p.371 of the 1956 edition
Please note that Apocryphal texts are not considered heretical within the Orthodox Church, they are considered the texts that are supposed to be read in private, instead of within the church.
At this point St. Palamas points out that our Most Pure Lady while dwelling in the temple had "unceasing holy prayer" (which is what Orthodox are supposed to attempt to do) At this point Mary was entirely focused on God.
I can continue on here, but that would be an entire separate topic. With many more quotes and references.
The Theotokos was sinless, but still tainted by her Human Nature. She was the best a Human could be, even though that is not enough she is still the most exemplary christian role model other than Christ Himself.
The Theotokos still needed a savior.
Concerning Pelagianism, the last of the "Great Heresies"
The heresie is concerned with the nature of humanity, with the regards to the fall and free will. Which ends up with the question, is salvation a matter of God's will or the free will of humanity? Does God merely make an offer of salvation that humans beings can choose to accept or reject, or does God actually save people?
The root of the error of Pelagius is an overly simplistic understanding of sin; in fact he had no idea of sin as such, only of sins, individual and separate acts of disobedience to God’s good Law. He taught that every man is created like Adam, free from sin and equally capable of choosing either good or evil. He denied such a thing as an inborn bent to sin, and attributed the observable fact that everyone sins to the force of bad example. Not only that, but he really attached no weight to the question of human moral character; yet this is a vital point, for our moral choices are not isolated acts of a will that is balanced between good and evil, but acts of a person who has a history, preferences, desires, and moral bent. If, on the other hand, sin is a matter of individual and separate acts of the will, and people sin because of bad examples around them, then virtue is also a matter of individual and separate acts of the will, guided by good examples.
This, in Pelagius’ teaching, was the function of the Scriptures. The Law tells us what we should do, and the Gospel shows us the good example of Christ’s obedience. It is ‘good news’ because it tells us that we can keep the law after all, not because it is the unveiling of a righteousness which is from God as a free gift. He even taught that some Biblical characters, like Daniel, had lived free from sin all their lives.
The inner logic of this false position was that it was not only possible, but absolutely necessary, for people to cease from sinning altogether. Having thus ceased from sin, a Christian would be acceptable to God in his own right.
While he said that Christians need the grace of God for salvation, in reality he redefined the grace of God to mean the free-will that God gave all people and the gift of the perfect moral law and example of Christ. The grace of God was in other words a matter of gifts that were common to all mankind.
This is not the same as the Orthodox understanding of Ancestral sin and the consequences of it.
It was St. Augustine who was the main opponent of Pelagius. Whereas Pelagius was one of those people who have very little sense of sin, and had been a monk from an early age, Augustine had the bitter memories of an early life filled with sin, and mourned that he had sought God so late in his life. The result of this was that he had a very vivid sense of the sinfulness of man and the corresponding greatness of God’s grace, while Pelagius did not; this is one of the reasons that Augustine’s Confessions is one of the great spiritual autobiographies.
"All good things come from you, O God, and from my God is my whole salvation," (St. Augustine, Confessions 1.6.7).
Pelagius saw all children as born innocent and pure... but Augustine wrote
"children are innocent only because they do not yet have any physical strength; their minds are not innocent" (Ibid 1.7.11)
This points to the ancestral sin taint on all of humanity. A most difficult thing to deal with. It is interesting to see how the Theotokos and Pelagius both were given ideal circumstances to avoid sin being given a religious "head start" and yet despite that Theotokos did not sin in thought, while Pelagius is an amazingly "great" heretic.
Anyways... most of the refutation of the Pelagius heresy is found in Romans. 3:10-19, 5:14-19, ... and other verses elsewhere.
You didn't really ask for a full history of it, so I'll stop here.
Short answer:
Orthodox Church: We are "infected" by the sins of our fathers and mothers, we also suffer the consequences of their sins, this is why we need to be cured/saved.
Catholic Church: We are guilty of the sins of our fathers and mothers, that is why we need saved.
Pelagius: We are not guilty of the sins or affected by the consequences of our fathers and mothers. We can become righteous to God.
The Theotokos would appear to fulfill the requirements of Pelagius, but She still needed a savior.
