No, I basically agree with you. However, neither the law of conscience or the law of Moses was the basis of universal death between Adam and Moses. But, there was the law in Genesis 2:17 that was the basis for universal death and Paul makes a very clear and repetitive case that was the law that brought universal death upon all mankind.
Here is what God told Adam, knowing redemption through the blood of Christ, as of a lamb without blemish and without spot, had been foreordained before Adam was created. Redemption from death.
Gen 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. I agree, the law.
And when the woman saw that the tree
was good for food, and that it
was pleasant to the eyes, and a tree to be desired to make
one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat. Gen 3:6
Did the transgression of that law result in?
Behold, I was brought forth in iniquity,
And in sin my mother conceived me. Psalms 51:5 NKJV
Who was the first, in iniquity brought forth, in sin conceived?
Here is what God told Joseph in a dream through an angel.
“Joseph, son of David, do not be afraid to take to you Mary your wife, for that which is conceived in her is of the Holy Spirit. “And she will bring forth a Son, and you shall call His name JESUS, for He will save His people from their sins.”
So all this was done that it might be fulfilled which was spoken by the Lord through the prophet, saying:
“Behold, the virgin shall be with child, and bear a Son, and they shall call His name Immanuel,”[fn] which is translated, “God with us.”
Did that have anything to do with being, without spot and without blemish? Is that the law under which Christ was born?
Joseph's response.
Then Joseph, being aroused from sleep,
did as the angel of the Lord
commanded him and took to him his wife, and
did not know her till she had brought forth her firstborn Son.[fn] And he called His name JESUS.
But when the fullness of the time had come, God sent forth His Son, born[fn] of a woman, born under the law, to redeem those who were under the law, that we might receive the adoption as sons.