Harav Sholom Schwadron, zl, cites a story that occurred concerning the Chafetz Chaim that gives insight to the Patriarch's decision.
The Chafetz Chaim permitted a student who was suspected of transgressing a number of sins to remain in the yeshivah. A short time later, a woman came to complain that one of the yeshivah students who ate his meals at her house was a mechutsaf, insolent. The Chafetz Chaim immediately had the student dismissed from the yeshivah.
When asked why he kept a student whose shortcomings amounted to much more serious sins than chutzpah, the Chafetz Chaim responded by citing the Mishnah in Avos 5:20 that says, "Az panim, the brazen-faced is headed for Gehinom; boshes panim, the one who is a shame-face will go to Gan Eden."
What is the Mishnah teaching us? If the az panim does not repent, he will certainly be punished; if he performs teshuvah, why should he not be accepted in Gan Eden? Furthermore, why does the Mishnah emphasize the "az-panim" in this case more so than any other sinner? Does not anyone who sins go to Gehinom? The Chafetz Chaim explained that while teshuvah ostensibly "works" for every sin, the chances are that one who is brazen will probably not repent. This is indicated by the text that stresses the az panim, one who is brazen, as opposed to azus panim, brazenness. The person who is brazen, who is steadfast in his brazenness, will not repent. In order to perform teshuvah, he must purge himself of brazenness and arrogance.
Veritably, the student whose sins go far beyond chutzpah might presently be faced with a formidable challenge in his quest for teshuvah. There is, however, hope. The one who is an az panim, whose brazenness is uncontrolled - will not repent. He will continue along his negative path, arrogantly doing whatever he wants, unconcerned with the people he is hurting.
Undoubtedly, if the "az panim" repents, his teshuvah will be accepted. Unfortunately, such a person rarely repents. He sees nothing wrong with what he is doing. Even if he were able to be introspective, his arrogance would never permit him to acknowledge the reality.
Amalek's most significant negative character trait is chutzpah. He is not "nispael", impressed, by anything. He fears no one and constantly demonstrates his insolence. The entire world feared Klal Yisrael when they left Egypt. Yet, Amalek was the first to challenge them. The Avos perceived in Timna, his mother, this tinge of "azus panim." They understood that one who possesses this middah will not repent; she will not be able to conform sincerely to Judaism. Thus, they did not accept her.