It is true that people often pull scripture out of context to say what it is not saying. Psa 51:5 is one such example. David is not discussing Original Sin in this Psalm, but confessing his personal sin with Bathsheba.
Psa 51:
1 To the chief Musician, A Psalm of David, when Nathan the prophet came unto him, after he had gone in to Bathsheba. Have mercy upon me, O God, according to thy lovingkindness: according unto the multitude of thy tender mercies blot out my transgressions.
2 Wash me throughly from mine iniquity, and cleanse me from my sin.
3 For I acknowledge my transgressions: and my sin is ever before me.
4 Against thee, thee only, have I sinned, and done this evil in thy sight: that thou mightest be justified when thou speakest, and be clear when thou judgest.
5 Behold, I was shapen in iniquity; and in sin did my mother conceive me.
It is clear David is confessing his own personal sin in this Psalm. But some would have us believe that in verse 5 David suddenly changes the subject to Original Sin and blames his mother for being born with a sin nature.
What David IS saying is that he was born of a Jewish mother who had been either married to a non Jew and was divorced (because Nahash was alive and showed mercy to David for his sisters' sake), or else had borne children to a non Jew out of wedlock. This was considered a pollution. David was scorned by his own brothers and his own father who did not present him when Samuel asked to see all of Jesse's sons.
Actually, in this David is a type of Christ. Jesus was also scorned by his half-brothers.
John 7:3 His brethren therefore said unto him, Depart hence, and go into Judaea, that thy disciples also may see the works that thou doest.
4 For there is no man that doeth any thing in secret, and he himself seeketh to be known openly. If thou do these things, shew thyself to the world.
5 For neither did his brethren believe in him.
Just as David was mistreated by his half-brothers because of his (in their view) less than legitimate birth, Jesus was mistreated and misjudged in the same way, his own family did not believe in him. David was also assigned to care for the sheep, a figure of Christ our Shepherd.
1 Sam 16:11 And Samuel said unto Jesse, Are here all thy children? And he said, There remaineth yet the youngest, and, behold, he keepeth the sheep. And Samuel said unto Jesse, Send and fetch him: for we will not sit down till he come hither.
If anything, this verse is teaching that David was unjustly considered sinful because of his birth. We do not bear the sins or iniquities of our parents.
Eze 18:20 The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.
But back to the OP, Psa 51:5 is a perfect example of pulling scripture out of context as a proof text to prove a presupposition that it did not address.
If Psa 51:5 were to prove we are born with a sin nature (which it does not), it is still a problem for OS, as it would say he received his sin nature from his mother, not his father.
Let's stop derailing the thread. It's about proof texting and not about original sin. Start your own thread. If you do, I have a good answer that may help you out with the passage, but I won't do it here as that is not what the thread is about.