Of the many malevolent characters recorded in the Bible, Amalek, the grandson of Esau, stands out as the archetypal enemy of our people. He is the father of the nation that waged war against the Jewish people upon their miraculous emergence from Egyptian bondage.
Throughout our long history, Amalek had many anti-Semitic heirs. One of these was Haman, in the Purim miracle, who sought to annihilate every Jew.
What are the origins of this character?
“Timna was a concubine to Elifaz, son of Esau, and she bore Amalek to Elifaz.” (Genesis 36:12)
Some verses later, we are told about Timna’s brother, Lotan. “These are the chiefs who came of the Horites: the chief Lotan, the chief Shobal, etc.” (Genesis 36:29)
Timna was the product of a royal family, but her brother held the position of one of the Horite chiefs.
The Talmud (Sanhedrin 99b) fills in missing details. Timna sought to convert to Judaism and came to Abraham, Isaac and Jacob, but they refused to accept her. Timna produced Amalek because they rejected her.
Timna became a concubine to Elifaz, insisting that “it is better to be a maidservant to this nation than to be a princess in any other nation.”
Being from the seed of Abraham and Sarah was extremely prestigious. Timna knew the great honor given to them and was willing to sacrifice her personal dignity to be a part of this prominent people, even if her connection would be a servile one.
I’ve always been puzzled by this story.
Timna’s actions seem righteous and altruistic. Why would this result in a child who represents the very paradigm of evil?
Our patriarchs and matriarchs selflessly reached out to humanity teaching G‑d’s path of truth. Abraham was the ultimate seeker of converts. If our patriarchs rejected Timna, they must have perceived that her motives were insincere.
It seems that Timna was not searching for G‑d, but personal aggrandizement. Had she truly wanted to join a nation serving G‑d, she would not have “settled” with the family of Esau, whose path was the antithesis of Abrahamitic teachings. Instead, she desired to be part of a powerful, but not necessarily G‑dly or spiritual, people.
Timna was willing to sacrifice her personal status, but only for greater, more fulfilling gratification. Her actions were not about G‑d, but entirely about herself. Our perceptive patriarchs realized this and rejected her.
In the era of Redemption, evil will be eradicated and all humankind will be rectified. Only one nation—Amalek—will not be a part of this vision.
“Their hand is on G‑d’s throne.” (Exodus 17:16)
Amalek represents an attitude that is beyond repair because it is the opposite of G‑dliness. It is entirely egocentric, even while dressed up as an act of devotion to G‑d. These are the seeds that can grow into the most extreme acts of evil fathomable.
Because the greatest distortion of goodness is an act of malevolence adorned in the religious garb of purity—an act of selfishness acted out in the noble name of G‑d.

Wow, what a great insight!
Hadn't thought of it like this, that Timna clearly sought honor and prestige without being remotely interested in connecting to Hashem or working on her character or other spiritual inner work.
We obviously made a mistake by not accepting her. There is a discussion in the Talmud about whether or not she initially wanted to marry into the family. The answer is that she did want to marry and was not accepted. She should have been allowed to convert anyway. We have been punished for a long time for this. The Midrash explains that the sar (spiritual image) of Amalek is Satan himself. HaShem did not allow the Bnai Yisrael to annihilate Amalek since no nation can be destroyed before its sar in Heaven has fallen first. He will perish only when sin is no longer to be found in K'lal Yisrael. Every time a Jew sins, they add strength to the sar of Amalek. It is only when the Bnai Yisrael will be free from sin that Amalek will be destroyed (Alshich). This from The Midrash Says Shemot B'shalach page 161. Shalom.
Furthermore:
~ A woman once came to Rabbi Eliezer to convert, and said to him: “Bring me
near.” He said to her, “List for me your deeds.” She said: “My youngest son is
born of my eldest son.” He admonished her. She went to Rabbi Joshua, and he
received her. His students said, “Rabbi Eliezer drove her away, and you draw
her near!” He said, “When she set her mind to convert, she was no longer of
this world, as it is said, All who come to it (to the faith) shall not return (to
sin; Prov. 2:19)." --- Midrash Kohelet Rabbah 1:25
~ Turn back, my daughters... (Ruth 1:12) Rabbi Shmuel bar Nachmani said in
the name of Rabbi Yudan the son of Rabbi Chanina: Three times it says, turn
back, corresponding to the three times one must rebuff one who seeks to
convert. But if he persists further, we receive him. But Rabbi Yitzchak said:
The stranger did not lodge in the street. (Job 31:32) Always push away
with the left hand, and draw near with the right. --- Midrash Ruth Rabbah
2:16
Finally, according to Book of Elijah, chapter 24, as pointed out by Rabbi Gefen elsewhere:
"[Amalek] asked [his father Eliphaz], 'Father, who will inherit This World and the World to Come?' 'The Children of Israel,' replied [Eliphaz]. 'Go out and dig wells for them and fix roads for them. If you do so, your share will be with the lowly among them and you will enter the World to Come.'"
Shalom Dersu,
I agree with your comments. The question is, did Amalek do this? It seems obvious that the original son of Eliphaz (Amalek) was only human and that he did eventually pass away. Therefore what the Sages are referring to is the spiritual aspect of evil that Amalek represents. This evil has not been erased at this time.
Thank You,
Rabbi Jay