No. It doesn't say that God sacrificed animals and used their blood as a sin covering. It just says God made them coats of skins and clothed them.
If a man made the coats of skins and clothed them then we could assume what you assume. But when God clothes us it can mean something entirely different.
I haven't decided for 100% that you are wrong. Your assumptions could be correct. I'm just not sure yet.
This view seemed plain to me, even though I had not heard it from others, it is one I have held for a long time
Since you questioned this, I have looked at several internet sites that agree with this view.
So I am not out there all by myself on this issue. I wanted to check if I was outside accepted thought on this.
I have quoted PART of one of them here:
One source for such a teaching is at
Sermoncentral.com.
"The First Sacrifice is recorded in Genesis 3:21.
Remember the setting? Adam and Eve sinned against God, their eyes were opened, they realized they were naked, and they were ashamed. So they tried to cover their nakedness with fig leaves. But fig leaves make lousy clothes. So God intervened, sacrificed an animal, and make garments of skin for them.
This is the first sacrifice mentioned in the Bible. An innocent animal, most likely lamb, lost it’s life so that Adam and Eve could be covered.
Thus, as with all of God's institutions,
it was established by God Himself."
This first sacrifice has the picture of Christ all over it. It was a covering for the sin, made by God. Most likely a firstborn animal.
I'm not holding firm to a lamb but I do like the idea.