No, read it again.
Paul states that where there is no law (itself), there is no transgression (sin is not taken into account, imputed).
He does not say where there is no knowledge of law.
Ro 5:12-21 is an involved argument showing that the sin of Adam is imputed to the sons of Adam,
which is what is called original sin.
Elin is following Augustine's teachings on this matter.
Nowhere does the text state that the guilt of Adam is passed to all his descendents via imputation.
Here is the text...
Rom 5:12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
Sin first entered the world through Adam, while it was Eve who first actually sinned Adam is simply representative because he was the head of the household. Adam was led by his wife instead of led by God. It was due to sin that death resulted and the death spoken of here is not physical death but spiritual death.
Remember God had warned, "they day you eat... ye shall surely die." Adam died the very moment he sinned and it was not physical. It works like this...
Jas 1:14 But every man is tempted, when he is drawn away of his own lust, and enticed.
Jas 1:15 Then when lust hath conceived, it bringeth forth sin: and sin, when it is finished, bringeth forth death.
It is lust which brings forth sin and sin when it is finished brings forth death.
Eve lusted...
Gen 3:6 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.
Her lust brought forth sin and in turn brought forth death. That is sins wages...
Rom 6:23 For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.
Sin will ALWAYS bring death. Rebellion to God kills us to God, it destroys the spiritual connection because we turn to our own way and even worse it brings with it condemnation for we are guilty.
Paul wrote this...
Rom 7:9 For I was alive without the law once: but when the commandment came, sin revived, and I died.
Rom 7:10 And the commandment, which was ordained to life, I found to be unto death.
Rom 7:11 For sin, taking occasion by the commandment, deceived me, and by it slew me.
That is how sin works. Sin requires a "knowledge of right doing" to kill because sin involves REBELLION. It is rebellion to God which kills an individual INSTANTLY.
That is the sin and death Adam brought into the world, and this death has passed onto all men why? Because ALL HAVE SINNED. What happened to Adam has happened to all of us when we CHOSE to sin.
We are all fully responsible for our own sin and resulting death. Original Sin denies this and puts the blame squarely on God who allows it to happen apart from any choice of an individual. Can you see how this maligns God's character? It is an evil doctrine, it is a dark doctrine, it is a deceptive doctrine which redefines everything to do with the Gospel.
Paul continues...
Rom 5:13 (For until the law sin was in the world: but sin is not imputed when there is no law.
Doing wrong was in the world without the law of Moses but sin is not imputed without law. Why? Because condemnation is only wrought through the exercise of free moral agency to choose evil. If it wasn't that way then God would be a vindictive tyrant punishing innocent people for lacking knowledge. God is not unjust.
This is why Paul in Romans 7 teaches that he was alive once without the law. When Paul did not know right from wrong (like a baby) he simply could not sin. Sin takes occasion by a commandment or knowledge. Now that knowledge can either be a direct command given by God like the Decalogue or it can be knowledge wrought by our conscience (which is the light of God that appears to all men). This is why a child can only sin unto death when it reaches an age where it has the mental capacity to understand right from wrong and therefore gains the ability to CHOOSE wrong.
So when Paul rights...
Rom 5:14 Nevertheless death reigned from Adam to Moses, even over them that had not sinned after the similitude of Adam's transgression, who is the figure of him that was to come.
... he is speaking that death (spiritual) reigned in the world because all sinned whether it was a violation of law (like Adam and after Moses) or a violation of conscience (between Adam and Moses) where they sinned not in the same way as Adam.
How can someone read an inherited sin nature into those verses? I'll tell you how.
AUGUSTINE OF HIPPO.
Augustine was the Bishop of Hippo around 400AD and he was steeped in Gnosticism before he converted to Catholicism. It was Augustine who developed the Original Sin doctrine and he did it by using the Latin Vulgate which says "in whom" instead of "that all have" in the Greek. Augustine thought that all the souls of humanity were present in Adam and thus sinned when Adam sinned. This teaching has a fancy theological term called "Traducianism" which simply is an allusion to how souls are birthed. Augustine viewed that all souls come into this world already condemned and corrupted because of this doctrine he developed.
Augustine was schooled in "dual nature" doctrine by Neo-platonist and Manichaean instructors and literature. "Dual nature" doctrine teaches that matter itself is evil and that the flesh body entraps a soul and prevents it from being virtuous. Augustine moulded this teaching with Christian terminology and was able to get it accepted as orthodox due to his prolific influence and writing ability.
Both Martin Luther and John Calvin were heavily influenced by the works of Augustine and both of them brought the Original Sin teaching in as the foundation for their doctrine. Martin Luther wrote "Bondage of the Will" where he teaches that the human will is bonded from birth to sin due to the fall of Adam whilst John Calvin taught total depravity. Their whole theological system is premised off this foundation and it permeates EVERYTHING established on top of it. It totally redefines the Gospel message to a "saved IN sin" message instead of a "saved FROM sin" message the multitudes of people who buy into it are going to be rejected at the judgement because they refused to repent and approach God in the proper manner, instead they bought into an approach that was a fairy tale.