The Torah teaches us (Vayikra 22:27) that from the 8th day, a bullock, sheep or goat is accepted as a offering for G-d:
When a bullock, or a sheep, or a goat, is brought forth, then it shall be seven days under its dam; and from the eighth day and thenceforth it shall be accepted for an offering made by fire to the Lord.
The Midrash (Pirkei DeRabbi Eliezer, chapter 29) connects this idea to the fact that Isaac was circumcised on the 8th day, thus, says the Midrash:
Thou mayest learn that everyone who brings his son for circumcision is as though (he were) a high priest bringing his meal offering and his drink offering upon the top of the altar.
Similary, on Bereishsis 17:13, the Rabbeinu Bahya explains:
A Midrashic approach to our verse views the commandment as similar to the offering of a sacrifice. Just as when a person offers a sacrifice on the altar and the blood of the offering achieves atonement for him, so the blood lost by the person being circumcised acts as atonement for him. This is why this rite has to be performed on the eighth day of the baby’s life. Just as an animal sacrifice is not admissible as such until the eighth day of the animal’s life (compare Leviticus 22, 27: ומיום השמיני והלאה ירצה, “and from the eighth day on it will be welcome as a sacrifice for G-d. In connection with the sacrificial offering the Torah writes (Exodus 29,33) ואכלו אותם אשר כפר בהם, “They- who received atonement through them (the offerings)- shall eat them;” so we make a festive meal to celebrate the induction of the circumcised baby into the Jewish community. In fact, the effect of the circumcision is far greater than that of an animal sacrifice. An animal sacrifice represents only a financial sacrifice by the donor whereas the rite of circumcision involves his body. The organ which is cut is one that radiates feelings to all parts of the body of the person concerned. This is why it is called ראש הגויה, “head of the whole body” in Negaim 6,7. This is why this organ was chosen to serve as a demonstration of self-sacrifice and why someone who undergoes circumcision is considered in the eyes of G-d as if he had sacrificed his entire body on G-d’s behalf. This is also why in Psalms 50,5 the psalmist Assaph describes circumcision as “who made a covenant with Me over sacrifice.” G-d Himself describes His pious as having made a covenant with Him by means of circumcising themselves.
So, according to these sources, the reason why we wait 8 days, is because it is considered as if you bring a offering, as the Pirkei DeRabbi Eliezer explains.
See also the explanation given by the Lubavitcher Rebbe (Likkutei Sichos, volume 3, p. 834).
You ask, "This doesnt answer why a child has to be in such a disgusting state for seven days"- to this, I would like to add in the commentary of the Rambam in his Moreh Nevuchim (3:49) that the reason we wait 8 days, is because it is not safe for the baby to give his bris milah before the 8th day:
The circumcision must take place on the eighth day (Lev. 12:3), because all living beings are after birth, within the first seven days, very weak and exceedingly tender, as if they were still in the womb of their mother; not until the eighth day can they be counted among those that enjoy the light of the world. That this is also the case with beasts may be inferred from the words of Scripture: "Seven days shall it be under the dam" (Lev. 22:27), as if it had no vitality before the end of that period. In the same manner man is circumcised after the completion of seven days. The period has been fixed, and has not been left to everybody's judgment.
So, "This doesnt answer why a child has to be in such a disgusting state for seven days"- in this case it seems that the concept of pikuach nefesh is obviously more important. It is not safe to perform the Bris before the eighth day according to the Rambam.