For on this day, [G-d] will effect atonement for you in order to purify you.
Rather than "surgically" removing our misdeeds from us, Yom Kippur removes us from our misdeeds, by elevating us far above them. Therefore, the focus on Yom Kippur is on our relationship with G-d rather then on our misdeeds per se. What is required of us on this day is to yearn to be reconciled with G-d in a general sense, and to express this yearning by observing Yom Kippur properly.
Yom Kippur elevates us this way because the day itself reveals the intrinsic connection that every Jew shares with G-d by virtue of his or her Divine soul. The connection between our essence and G-d's essence has existed since before Creation and is therefore not limited by time or space. For this reason, it cannot be damaged by any misdeeds we might have committed. Thus, the very day of Yom Kippur - by revealing this intrinsic connection between ourselves and G-d - wipes our slate completely clean.