Why did we need goralos on the seirim,why a goral?
You have asked a very important and basic question. Here's what we have come up with, let us know what you think!
The reason for the unusual manner of choosing the goats on Yom Kipur touches upon the fundamentals of Jewish belief. We know that the basis of Emunah is the belief in Hash-m's individual supervision of all of our actions and that nothing in the world outside of Hash-m's will. It is that belief which is the basis for the belief in reward and punishment and for our obligation to fulfill the Mitzvos.
On Yom Kipur, the entire nation repents and asks Hash-m to forgive the sins of the past year. The most important part of this process of Teshuvah is strengthening one's Emunah in Hash-m. This is done by realizing that every single act that one did was under the watchful eye of Hash-m, and that everything that occurs to us was orchestrated by Hash-m.
It is the Goral that demonstrates most powerfully this "Hashgachah Pratis" of Hash-m in every detail of the world's existence. When a Goral is conducted, one has no input or influence on the outcome of the Goral. By choosing the Se'irim in this manner, we are proclaiming our belief that Hash-m, and not "chance," determines the outcome of seemingly random acts. By relying on the outcome of the Goral we show that our every experience is a direct result of Hash-m's supervision of what occurs. A Goral exemplifies that what seems entirely random, actually demonstrates the exclusive involvement of Hash-m.
In fact, we find that the Hashgachah of Hash-m was shown by the Goral with particular clarity. During the era in which the Jewish people were found deserving, the Goral la'Shem always came up in the Kohen Gadol's right hand, and when they were not deserving it did not (Yoma 39b). The people's belief that Hash-m watches over them at all times was strengthened through the Goral.
Since it is these Korbanos -- the Se'ir la'Shem and the Se'ir la'Azazel -- which effect atonement for the Jewish people, it is fitting that they be selected in a manner which engenders this most important element of Teshuvah.
II
Whenever the Jews weaken in their belief that Hash-m is watching closely their every action, Hash-m immediately sends Amalek to "wake them up" (Rashi to Shemos 16:8). The verse describes how Amalek attacked the Jews after they were redeemed from Mitzrayim and says "Asher Karcha ba'Derech" (Devarim 25:18). RASHI there says that "Karcha" means "they chanced upon you" (from the word "Mikreh"). Amalek set for themselves the goal to convince the world that everything is random and left to chance. They attacked the Jewish nation as they left Mitzrayim to show that the good fortune of being freed from bondage was merely happenstance; one day they experience redemption, the next day war -- it is all by chance. The only way the Jewish people were -- and are -- able to defeat Amalek is by directing their attention to Hash-m and to the belief that He, and only He, controls everything in the world. This is the meaning of the verse, describing Moshe's actions during the war against Amalek, "And [Moshe's raised] hands were Emunah" (Shemos 17:1) his arms upraised in prayer were a source of Emunah (see Mishnah Rosh ha'Shanah 29a). When we realize that everything is directed by Hash-m, we successfully defeat the forces of Amalek.
Amalek is the leading grandson of Esav, whose guardian angel is none other than the Satan himself ("Sama'el" -- Rashi, Sukkah 29a and Sota 10b, from Tanchuma, Vayishlach 8 -- Midrash Raba at the end of Devarim identifies Sama'el with the Satan and the Angel of Death). The Se'ir la'Azazel, according to the Midrash (Ramban, Vayikra 16:8) is meant as an offering to "appease the Satan." We silence the Satan the same way we silence Amalek, Satan's people -- by showing that there is no Mikreh and that everything is directed by the hand of Hash-m. We choose the Se'ir la'Azazel specifically by way of a Goral, thereby showing our belief that "chance" is orchestrated by the active guidance of Hash-m.
III
Chazal tell us that Haman is descended from Amalek (see Rabeinu Bachyei Shemos 16:17). When Haman plotted against the Jewish people in the times of Purim, he used the same strategy that Amalek had used centuries earlier. He conducted a Goral, or a "Pur," in order to choose a day on which to attack the Jews. He chose to use a Goral in order to show that the force of nature, of chance, is powerful enough to destroy the Jewish People.
Mordechai realized Haman's intentions. He told to Esther "Kol Asher Karahu" (Esther 4:7). The Midrash says that Mordechai was telling Esther that "the one about whom it is written 'Ashar Karcha ba'Derech' is plotting to attack us!" Mordechai described Haman this way to show that Haman was taking advantage of the fact that the Jews of the time had weakened in their Emunah in the Hashgachah of Hash-m. Haman -- i.e. Amalek -- used that as an opportunity to attack them. The response to such a threat is to regain the realization that Hash-m is involved with everything in the world and to recognize the hidden hand of Hash-m even in a time of "Hester Panim," a time of "Haster Astir Panai ba'Yom ha'Hu" (Devarim 31:18, Chulin 139b). The Jews had to overcome their lack of Emunah and realize Hash-m's hand in the world.
This is alluded to in the Megilah, which was purposely written without any mention of the name of Hash-m. That shows that even when the hand of Hash-m is not openly apparent, it is nevertheless active in the world, albeit hidden.
This is why the festival was named "Purim," in recognition of the "Pur" which Haman conducted (Esther 9:26). The Jews realized that the real cause of their troubles was their weakness in Emunah and their mistake in thinking that there is such a thing as Mikreh, chance, as represented by a Pur or Goral. Their victory was assured when they did Teshuvah and came to the realization that even a Goral is directed by Hash-m. In that sense, "the Pur of Haman was turned into our Pur " (from the Tefilah after the reading of the Megilah). The forces of randomness were shown to be non-existent, and everything was indeed seen to be guided by Hash-m.
IV
The Tikunei Zohar calls Yom ha'Kipurim as a "day which is like Purim" ("Yom k'Purim"). In what way Yom Kipur similar to Purim? The common theme of both days is the understanding that everything is determined by Hash-m, even the seemingly random outcome of a Goral and Pur.