The point of the OT law was to ultimately show the need for Christ. You see it is impossible for humans to fulfill or follow throughout their life the law of the OT. That is why there were sacrifices in the OT, to sacrifice something for their sin. Ther sin is lawlessness as said in the NT, and thus this grand law was to show the need for a Messiah that would fulfill it or live by it within His life to every letter. Christ did that, and thus it shows through OT context He is the Son of God.
The OT Law is not to show a need for Christ. Yes, it is not possible to fully follow the law at all times. And neither does God expect one to do so. That wasn't the point of the law.
Yes, there were sacrifices prescribed through the law, but that was a small part. Even today, the law is practiced, and there are no animal sacrifices. There are other ways for atonement.
The sin wasn't lawlessness though. The NT may say that, but it is wrong. There were many sins described in the NT. It was doing evil in the eye of God (the book of Judges is a great example of this).
The need for the Messiah had nothing to do with the Law though. It had nothing to do with fulfilling the law. The Law, for Jews, was not something to be abolished. It was a blessing. It was a gift. And they followed the law out of love for God. God commanded it, and they followed it out of love.
The need for the Messiah began after the Jewish people were conquered. The Messiah was suppose to save the Jews from foreign rule, and make the a great nation once again. When the law was first handed down, there was no need for the Messiah. It was later, when the people were conquered, that the need for the Messiah arose. It had nothing to do with abolishing the law, or fulfilling the law.
Finally, Jesus did not obey the complete law. On one occasion, he directly taught against the law (in the case of divorce). On another occasion, he broke the law by disrespecting his mother.