Amen Endure!
Often we believe that we are essentially good inside. We make mistakes because we are not perfect, but they are just honest mistakes. We have lost much of our childhood innocence, but if we are diligent, we can regain this. Intrepid, as you recognize, this is a lie.
As Endure said well, we are born with a very nature that is self-seeking and does not know the Lord. We are not children of the Lord, and He is not our Father. This is the inheritance, and as you are seeing and I am seeing, the concept of 'sin' is not so much doing something wrong as it is the depths at which we love to do wrong. We desire passionately to do the very things that in our mind we know to be false / lies / wrongs / unloving / etc.
However, it was not involuntary. Paul describes Adam's transgression as a type for all people, and so Adam's death came to all people "because all sinned" (12). The first question that comes to my mind is 'how could all haved sinned when only adam (Eve of course is implicated) was the one sinning?' I can certainly recognize the fact that I sinned in my life, but to have sinned in the Garden of Eden - I wasn't even there! How could my condemnation have been spelled out then, when I wasn't even born?
But I begin to see that it is because Adam is
typical of humanity - that is, he is the type head of all people. And so when he willingly fell, all willingly fell with him. In our culture especially, the concept of all people as one is completely foreign, but the Bible again and again speaks of people's participation in actions and deeds (the Cross is the ultimate example - all people crucified Christ, Christ's people have died with Him, and also are raised with Him - in the very next chapter 6 of romans

). So Adam's fall is indicative of our
active participation in the fall. We testify to this by our very nature, as we at one point or another actively push the Lord's ways from our ways.
"But the gift is not like the trespass" Praise God for this!
By Adam's one act of sin, all sinned, and all died in sin. But by Christ's "one act of righteousness" (18) is now justification for all of His people! As deep as sin goes, deeper is the gift. As far reaching as sin goes, further is the gift. And so now we have been brought to fellowship with the living Lord, who now calls us 'sons' and 'daughters', even as He calls Jesus 'son', and we call Him 'Father' because He has declared this before the whole world. Man, that's just sooo powerful to me.
So, we are all born with sin. This is infused from the beginning. We participated in Adam's sin, by our union with the type head of fallen humanity, so it is not involuntarily.
Now, accepting Christ as Savior and believing in Him for the gift of salvation is heriditary, not according to the flesh, but according to the promise (as Endure stated). That is, the children of God are children according to the Spirit. These are not children of birth in the flesh, but of rebirth in the Spirit. These that are united to Christ (as the type head of the new humanity, reconciled to our God) die with Christ and then live with Him and are worthy to be called children of the Living God according to the promise, because they have been declared in Christ and by the Holy Spirit to be His children.
It makes sense too why one couldn't be born a Christian - if natural birth declared salvation (Christianity), then the declaration would no longer be by the sovereign will of the Lord, but by the fallen will of the flesh. And that's also why our Lord says, "Jacob I have loved, but Esau I have hated", even though Esau deserved to be given Abraham's promise (passed to Isaac), since the physical human tradition is to pass the birthright to the oldest son. The promise is made, not by people, but by the living Lord. He makes the wise foolish, and the strong weak, so that all will know who it is that brings life.
Intrepid, I'm so glad you asked this - I too have been working this out along with you (sincerely desiring His ways, and not my ways). May we continue to seek our Lord in all of His Truth and wisdom, and live for Him.
-Chris