It is a very good point to understand the importance of blood. Just like you said, the significance of blood was to cover our sins so that we could be washed in righteousness through the precious blood of Jesus!

This is where we get the term "scapegoat" from as our common lingo. The priest laid hands on the goat without blemish and laid the other hand on the person who needed to be cleansed. A transfer of purity from the animal would transfer with the sin of the person and so then the goat was slain to complete the sacrifice.
Shedding blood was a yearly process during the day of Atonement but now we rest in the everflowing blood of Jesus which gives us access to righteousness when we have a heart of repentance!