There is no logic to applying an atonement before it takes place. The comparison to Yom Kippur and a sacrificial Passover lamb do not apply because their effect under the law did not apply in advance either. So if from the time of Adam people got salvation directly from God, then it can always work the same way with the eternal God. And Christian scriptures don't explain the alternative.
NT Bible is for those believed to be saved. NT Bible has also the purpose of filtering those refused to believe. These are the two main purpose of the Bible. Jesus ever said, those who believe will be given the abundance while those disbelieve even the last bit will be removed from them. The purpose of the Bible inline with what He said.
That being said. The OT is more like a preparation for Christ first coming. The preparation was done through the Jews. Such that it can be witnessed that how a flawless lamb being slain as an atonement of sin. This is rather an original Jewish concept. Such a concept was deliberately built via the Jews as a preparation of Christ's arrival. To put it another way, if God did't build the concept through the Jews. Jesus came then jumped off a cliff, or hanged Himself then died, if so how can humans know that the Son of God has been sacrificed?
Do you use credit card? Jesus blood doesn't need to take effect only after His death. It's rather the effect of His blood only explained (instead of effective) after His death. His blood shed takes effect long before He visited earth the first time, though this effect was only explained after His first visit. While the explanation is more focused on the salvation of humans today.
Do you study laws? Do you know how long it takes for a novice to study laws to become a professional lawyer? God doesn't demand humans to understand His Law and covenants in full details simply because it's never a requirement for humans to be saved. They only need to believe in Jesus Christ, for Christ sake!
Do you need to study laws in details in order to abide by laws? You can be a good citizen without studying the law details. You may need to watch daily news to get an update on here and there though.
That being said, I can highlight you the big picture here;
God has a full set of Law which is applicable to both humans and angels. Genesis is about how Satan the angel and Adam the man broke this set of Law. This set of Law is required to quality both angels and humans to enter the future Heaven (an eternity).
Adam was driven out of Eden (lied inside God's realm). Since then humans are living on earth (lying outside God's realm).
From Adam till Noah, it's a period standing witness that humans can no longer abide by this set of Law (which is applicable to both humans and angels). It is thus time to put an end to humans as God's purpose of bringing humans to Heaven is defeated. That's why the flood.
However God has Jesus Christ. By the effect of His blood, a counter reset can thus be called for. God can thus grant covenants to humans for them to be saved through Jesus Christ.
Covenants are the use of Christ Blood as God's Grace, such that humans can be saved alternatively. Covenants are only applicable to humans but not angels as the blood of Jesus is only for humans but not angels.
However as time goes by, humans are expected to go further and further away from God to an extent that they can't even stick to the covenant granted. To put it another way, the main purpose of a covenant is to distinguish the righteous (or saved) from the wicked (unsaved). If this purpose is defeated, it simply means a new covenant is needed. This is how it works even before Jesus' first coming.
Covenant thus has a time period for its this purpose to be served. It also has a scope of human coverage. Obviously, the Mosaic covenant is for the Jews (and converts) while Noah's covenant is supposed to be applicable to all mankind. It is also possible that today's humans (as gentiles) are under the coverage of this Noah covenant. Regardless, covenants also have the accumulative effect.
A covenant in effect only changes the way how one is judged. That's why even Christians need to deal with their own sins. It means they need to abide by whatever laws specified by a previous covenant though they are expected to failed here and there. The New Covenant on the other hand will protect them from being judged (on the Judgment Day) using those older laws. They will be exempted from such a judgment.
To view from another perspective, Grace of a covenant is not earned by humans. It is earned by Jesus' self-sacrifice. It is thus more like a gift. The one who earned it has the ultimate right to grant it to whoever He wishes to. It means that don't complain if the gift is not around you. All humans on earth are at least covered by the oldest one brought by Noah.
Preaching of the gospel (New Covenant) is an human effort. It is because humans have the right either to accept it or to reject it. It means God won't force it upon you if you choose to reject it. To put it another way, if it's not because at some point humans chose to reject it, it should have reached every corners of our world.
If you read the book of Revelation, you may notice that the final judgment is two-layered, namely, a judgment of covenant and a judgment of Law. In the judgment of covenant, when Jesus says that you passed then you enter Heaven, or else you are condemned. It is so because covenants are signed by His blood.
If you are disqualified by the covenant you are subject to (Jesus will be the Judge), then you are put to death under the judgment of Law (the set applicable to both humans and angels). It can be as simple as pointing out one of your sins, then you are done. It is expected that not a single human can pass this judgment of Law.
All left is what covers the scope of humans from Adam to Noah, as there seems not to be a valid covenant there. The Bible hinted that somehow they are covered by the New Covenant which is the same one covering today's Christians. The Bible however didn't put many details on how it is so though.
