I'll take a shot at this:

• The entire sacrificial system in the OT points towards Jesus. There's nothing inherently salvific about the shedding of animal blood; it's the just the way God chose to get the point across to humanity that sin yields a destructive effect that is costly to reverse. In other words, the OT practice of animal sacrifice just helps to bring about a consciousness of sin's weight, namely, by having the person who has committed the sin give something they value to God--usually by killing that thing. (There are offerings of things like grain or precious metals in the OT and not just animals.) It is always God doing the forgiving and reconciling; animal sacrifice is ultimately a teaching tool to get this point across.

  • "So then, the law was our guardian until Christ came, in order that we might be justified by faith. But now that faith has come, we are no longer under a guardian." - Galatians 3:24-25
    
  • "The Law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship." - Hebrews 10:1
  • "It is impossible for the blood of bulls and goats to take away sins." - Hebrews 10:4
  • "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." - Romans 3:20

• Jesus is not just a human sacrifice. Rather, Jesus can be understood as God more intensively expressing how forgiveness works, leveraging that sacrificial system from the OT with Jesus not only being like the animal sacrificed but also like the priest offering the sacrifice and also God accepting the sacrifice. Nothing much has changed here with respect to the primary agents in the process of salvation since it is still God doing the forgiving and reconciling of people who need that. But once humanity has (presumably) understood the destructive effect of sin and its costliness to reverse thanks to all that animal sacrifice stuff, we'll be able to better understand and value the forgiveness God offers as that is depicted by Jesus's sacrifice.

  • "The next day John saw Jesus coming toward him and said, 'Look, the Lamb of God, who takes away the sin of the world!" - John 1:29
  • "Therefore, since we have a great high priest who has ascended into heaven, Jesus the Son of God, let us hold firmly to the faith we profess." - Hebrews 4;14
  • "Unlike the other high priests, he [i.e. Jesus] does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself." - Hebrews 7:27
  • "In Christ God was reconciling the world to himself, not counting their trespasses against them, and entrusting to us the message of reconciliation." - 2 Corinthians 5:19

• Side note: Why not skip the whole animal sacrifice thing and go straight to God doing some Jesus-y stuff right off the bat? Basically, because God wanted to reveal this saving / reconciling / forgiving part of God's self to us. If we treat God as a transcendent being who exists beyond and yet also created material reality, including time, then it is within God's power to instantaneously reverse the damaging effects of anybody's sin; however, if God did that, we wouldn't know about it. It would be like an adult to always stops a kid from touching the sharp thing such that the kid never learns the damage that cutting themselves inflicts nor values the work the adult does to heal the kid with bandages and ointment and kisses and stuff. So, there's a sort of necessarily, pedagogically laborious aspect to the precise timing of Jesus's physical arrival on planet earth, as preceded by all the animal sacrifice stuff. Also, Jesus's sacrifice is temporally universal in that it not only affects people after Jesus arrived but also "traces back through time" as we understand it to affect people who lived before Jesus physically arrived.

  • "Blessed be the God and Father of our Lord Jesus Christ, who has blessed us in Christ with every spiritual blessing in the heavenly places, even as he chose us in him before the foundation of the world, that we should be holy and blameless before him. In love he predestined us for adoption as sons through Jesus Christ, according to the purpose of his will, to the praise of his glorious grace, with which he has blessed us in the Beloved. In him we have redemption through his blood, the forgiveness of our trespasses, according to the riches of his grace, which he lavished upon us, in all wisdom and insight making known to us the mystery of his will, according to his purpose, which he set forth in Christ as a plan for the fullness of time, to unite all things in him, things in heaven and things on earth." - Ephesians 1:3-10
    

• Finally, the "no human sacrifices" mantra in the OT exists in the context of the OT to prevent people from killing something God doesn't want to be killed in people's zealous attempts to achieve forgiveness. A particularly heinous aspect of several of the people groups surrounding the Israelites was child sacrifice, and some of the rulers of the Israelites who are depicted as being the most evil also did this, like Ahaz as discussed in 2 Kings 16:1-3. Nevertheless, the story of God commanding Abraham to sacrifice Isaac only to stop Abraham from doing so is probably the most famous narrative portion unpacking this prohibition as mentioned in Genesis 22. Christian theologians tend to read this story also as a foreshadowing of Jesus, expect that where Abraham was prevented from sacrificing his son, God actually went through with it in Jesus. Again, this only works because Jesus is functioning on all sides of the sacrifice--as the thing sacrificed, as the one conducting the sacrifice, and as the one accomplishing forgiveness through the sacrifice.

Edit: Accidentally hit submit before completing the last part.