One indicator that the Levitical sacrifices were insufficient is that they had to be offered each year for the nation and more often for individuals, depending really on the frequency of offenses. In essence, they were only effective for past offenses. The next time a sin was committed, the person and nation were susceptible to judgment if a sacrifice was not offered at the appropriate time.
Furthermore, there was nothing in the sacrificial system for some sins, including adultery and murder.
The Levitical Law only existed from the time of Moses as well, while Abraham lived 400-500 years before that. "Abraham believed God and it was credited to him as righteousness." Salvation has always been through faith in God's provision rather than through our own works.
Adam and Eve's attempts to cover their own shame with the fig leaves in the garden were insufficient. They needed the clothing that God provided (and the promise of a Deliverer who would crush the work of Satan that brought death to the world).
In Isaiah, God actually tells the nation of Israel to "stop bringing meaningless sacrifices". The issue was that they were "honoring God with their mouths while their hearts were far from Him." Dutifully carrying out rituals was never a means of salvation.
Judaism may believe that God no longer requires animal sacrifices and that there is forgiveness apart from those and apart from Jesus, but the Bible is very clear that this is not the case. The sacrifices were an act of faith in God, and the sacrifice of Jesus is the only thing that has ever provided atonement for sins, "For it is impossible for the blood of bulls and goats to take away sins." (Hebrews 10:14)
There is no other name under heaven given to men by which we must be saved. (Acts 4:12) Christianity is very clear that Judaism without Christ cannot provide forgiveness of sins. In fact, nothing else can provide forgiveness of sins.
Romans 3 states the following:
Now we know that whatever the law says, it says to those who are under
the law, so that every mouth may be silenced and the whole world held
accountable to God. 20 Therefore no one will be declared righteous in
God’s sight by the works of the law; rather, through the law we become
conscious of our sin.
21 But now apart from the law the righteousness of God has been made
known, to which the Law and the Prophets testify. 22 This
righteousness is given through faith in Jesus Christ to all who
believe. There is no difference between Jew and Gentile, 23 for all
have sinned and fall short of the glory of God, 24 and all are
justified freely by his grace through the redemption that came by
Christ Jesus. 25 God presented Christ as a sacrifice of atonement,
through the shedding of his blood—to be received by faith. He did this
to demonstrate his righteousness, because in his forbearance he had
left the sins committed beforehand unpunished— 26 he did it to
demonstrate his righteousness at the present time, so as to be just
and the one who justifies those who have faith in Jesus.
Note that it says, "He had left the sins committed beforehand unpunished". God's justice requires that all sins be punished. No sin was punished until Jesus died and endured the righteous punishment of God for sins.
So, neither Levitical Law nor good deeds nor being religious can achieve forgiveness of sins. Forgiveness comes by faith in Christ and in His work of atonement.