>Why can't Christ simply finish the work of salvation at the cross as OP said?
He did, of course, but, as Christ said, salvation is contingent on the repentance of the individual sinner. It helps to take the concept of sin as presented in the grand scale of the Gospel and to place it in a smaller, more tangible context. Let us say that you assassinated a princess; by even the loosest standards of morality that is a crime worthy of death. The prince of the kingdom firmly insists on taking the place of you on the execution block, and as bizarre and unprecedented as this seems to everyone else the King, who's came up with the idea in the first place, is only too happy to oblige. Do you:
A.) Conclude that no matter how many princesses you murder the King will instantly pardon you, and as such you don't even need to apologize for him to pardon you.
B.) Do the only thing you can and pour out your gratefulness and repentance to the King in a letter because, for some reason, the King, who lives too far away to meet you in person, did not establish a court system for criminals to be pardoned at all, or any centralized government at all for that matter.
C.) Show up to court, confess to the royal judge, and receive pardon.
Both B and C are perfectly viable ways of apologizing to the King. But if we live in the world where the King always relied on B in order to deal with criminals and never ever established C, the superior option, then his status as the only truly loving king in the world contrasts with the evidence that his kingdom is actually a splintered confederation that dosen't even have a judiciary system. If we live in the world where the judges weren't just corrupt, as government officials tend to be, but a false government entirely, which the King allowed to reign in his stead since the establishment of the kingdom (confederation), then the contrast is twice as stark.
Christ didn't establish confession as a sacrament because repentance as a concept is contingent on it; it isn't, and the OT shows that clearly. He established it, as John 20 shows, because if the when the King who is spirit decided to come in the flesh, he decided, like any good King would, to set up a proper government.