There are several problems in this statement, and I point them out simply for edification purposes--and to make you think and interact more deeply with the text.
You state: "Abraham, and everyone who made animal sacrifices"...had to do so yearly...
This is not necessarily so. You are viewing Abraham anachronistically through the Law of Exodus, Leviticus, etc. We know Abraham sacrificed, that's pretty much without debate. However, we don't see his sacrifices bringing "righteousness" to him. We clearly see Abraham's faith being counted as righteousness.
Now, in the performance of the sacrificial ritual under the Law, it was not understood that the sins were merely "pushed forward" for another year. Indeed, the text says that "atonement" was made.
Of course, we learn later--In Hebrews 10--that the blood of bulls and goats cannot and do not take a way sin. However, we must still deal with the Old Testament saying that "atonement" was made. (More on this below).
You state: "They were looking to the coming Messiah...whereas, we look back."
This is a fairly common misconception, although there are elements of this thought line that are not wrong. However, the thought line suggests that Abraham was trusting in Christ for his salvation, as were Moses, David, Daniel, etc. None of these men, or their contemporaries, were trusting in a God-Man from Nazareth named Jesus. Rather, all these men, were trusting in God--Yahweh--and living their lives according to His word. For Abraham this was trusting in the promise that God gave him. For Moses and David, this trust was expressed through right, lawful living while making "atonement" for their sins through the sacrificial system.
Now, how does all this fit together? How is "atonement" made in the Old Testament while Hebrews says the blood of bulls and goats can't take away sin?
Simply put: CHRIST. OK, we can all say "well, DUH!" now. But, here's how this works, according to Romans 3:
But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it—22 the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: 23 for all have sinned and fall short of the glory of God, 24 and are justified by his grace as a gift, through the redemption that is in Christ Jesus, 25 whom God put forward as a propitiation by his blood, to be received by faith. This was to show God's righteousness, because in his divine forbearance he had passed over former sins. 26 It was to show his righteousness at the present time, so that he might be just and the justifier of the one who has faith in Jesus. (Romans 3:21-26, ESV)
We're all familiar with this passage. But, the answer to the dilemma at hand is in verse 25 and following.
What we see here is that God put forward Christ as a propitiation for one major purpose--to show His [God's] righteousness. Why would He need to do so? Because He passed over former sins in His divine forbearance.
Now, what does that mean?! It means that God counted the blood of bulls and goats as "atonement" and that He counted Abraham's faith as righteousness without punishing their sin. In other words, Abraham, Moses, David, etc. were never held to account for their sins. God counted the sins as forgiven, even though they weren't.
So, the
primary reason Jesus came was to pay for those sins that God passed over in the past. This was absolutely necessary because God simply cannot "write off" sins. He can't say, "that's OK, don't worry about it, I'll just let it go." If He were to do so, He would cease to be God. All sin is against God and God, in His righteousness and justice is "required" to punish sin. Therefore, Jesus is the ultimate payment for the "passed over" sins of the Old Testament.
It is as if God put the sins of the "Old Testament Saints" on His credit card and paid the bill in Christ on the cross. It is in this way, and only this way, the sacrifices of the Law could be said to make "atonement." God could count the sins as atoned for since He knew He'd make payment in Christ.
This expresses one inescapable truth: There is no injustice with God. Every sin
will be paid for--either by the sinner in hell for all eternity or in Christ on the cross.
Blessings,
The Archangel
NE was saved under the old covenant. They were still in their sins until Christ sacrificed Himself for us. Their was no remission of sins until Christ's blood was offered as the sacrifice for sins. This is theology 101. Even my 12yr old understands this basic truth. They did not inherit eternal life until they were made coheirs of the promise. And this did not happen until the New & eternal Covenant was in force by the blood, person, work, & word of Jesus Christ. There was no salvation until the New Covenant. To believe otherwise is in direct contradiction to Scripture & could reveal a profound misunderstanding of the relationship between the old & New Covenants. 
