Jesus could and did forgive sins before the cross. This took place in the OT or otherwise known as a time under the old/first covenant. How could he forgive sins without the shedding of blood?
What changed after the cross? (his death and shed blood ushered in the new covenant)
The Law provided atonement for sin as a complete system. Failing in one part one fails in all of it. So not only was blood required, but there were other means of purification--not just the use of water, but water used in a particular way. Everything was sanctified by the word of God such that doing any of these things becomes common, useless, and perhaps idolatrous if not done in the particular order.
So blood was indeed required in order to obtain atonement for individuals and for the nation. Forgiveness apart from these ceremonies was possible only because these ceremonies were being carried out. For example, a person may abuse his neighbor and thus commit sin against God. He may apologize to his neighbor and confess his repentance before God, and he will be forgiven. But he is forgiven within the context of national blessing only if he does it under auspices of the regular offerings made by the priest in accordance with God's Law.
Certainly people outside of Israel could be forgiven, in accordance with their conscience. But they did not have the promise of national blessing God afforded Israel when they remained in compliance with the Law. Becoming united with God under a covenant enabled a person to obtain a special spiritual relationship with God with its corresponding blessings.
The Law, however, was just a temporary means of forgiveness and could not, therefore, provide for Eternal Life. Just one sin would disqualify Israel form Eternal Life. And we know, by testimony of the Law, that Israel regularly sinned and therefore constantly needed covering by the regulations of the Law.
This is what Paul meant by the Law increasing the knowledge of sin. Obeying the Law was a form of Israel confessing that they were, by nature, sinful, and thus needed covering by regulations of the Law, including blood atonement. And apart from this Israel did not meet conditions of their covenant with God, and therefore could not obtain special national blessing in their individual and national life.
Christ offered his own blood, which God required in order to bring not just forgiveness for special blessing but forgiveness for Eternal Life to people. He did this not just for Israel, but for all people.
Without this legal act, God refused to grant Eternal Life to anybody. People who don't know anything about Christ will be judged according to their conscience. But those who embrace Christ's covenant of forgiveness and life will obtain in this life spiritual life that is abundant and directly leads to Eternal Life.
Apart from Christ, nobody obtains the kind of life that is of the same nature as Eternal Life. This is the difference between living good lives and living the life God wants us to live, the difference between being "born again" and merely doing good acts on occasion.
If we want to experience the virtues of this abundant life Christ died to give us we must embrace Christ as our blood atonement and as the source of our spiritual life. We follow him as the divine Christ. In doing so we obtain his atonement by default.