The idea of being born totally depraved is read into Romans 3.
Paul spends Romans 1 proving Gentiles are sinners and then Romans 2 that Jews are sinners. NOWHERE in these 2 chapters does Paul say men are sinners for how they are innately born with sin/sin nature/totally depraved. If such were the case men are innately born sinners/sin nature there would be no better context than Romans 1 and 2 for that idea to be mentioned but it is not found at all!
After proving all, both groups Gentiles and Jews are sinners, we know sinners are in need of justification which is Paul's theme.
In Romans 3 , Paul spends the first part of Romans 3 telling us what does NOT justify (the OT law of Moses) and the latter part of the chapter Paul tells us what does justify, that being faith.
The OT law (or any other law like it) that requires the work of perfect flawless law keeping cannot justify for the Jew could not keep it perfectly, (Galatians 3:10)..."Cursed is every one that continueth not in all things which are written in the book of the law to do them". That OT law did not even require faith, just doing it perfectly was required to be justified by it (Galatians 3:12)..."And the law is not of faith: but, The man that doeth them shall live in them." Since the Jew could not keep all of it perfectly, then "no man is justified by the law in the sight of God" (Galatians 3:11).
Since that OT law that was given to the Jew could not justify the Jew, then the Jew was no better than the Gentile..."What then? are we (Jews) better than they (Gentiles)? No, in no wise: for we have before proved both Jews and Gentiles, that they are all under sin;"
How do we know the Jew was not justified by the OT law? For in Romans 3:10-18 Paul quotes some passages from the OT law proving the Jew sinned therefore unable to be justified by that OT law.
Points:
1) those quotes from Romans 3:10-18 are being directly specifically at the Jew, not every individual universally. For after Paul quotes those OT passages Paul says in v19 "Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God.". Paul refers to all those OT verses cited as "law" and that OT law was only given to the Jew, hence those verses in 10-18 are only directed at the Jew. Paul is simply telling the Jew that your own law given to you says you are a sinner.
"Here (Romans 3) Paul cites six O.T. passages which prove his point. These were from the utterances entrusted to the Jewish nation. Someone might argue that such passages were referring to wicked Gentiles, in verse 19 Paul will answer that argument, by pointing out that the 'Law', spoke to those under it, i.e. these passages were talking about Jews!" Dunagan Comm.
2) the only inability found in this context is the inability of that OT law to justify the Jew. Nothing at all is said about the Jew being innately born totally depraved. The fact the Jew could not keep the law perfectly does not prove 'total inability'. God has never required perfect flawless law keeping for one to be justified, but God does require a simple faithful obedience which man IS CAPABLE of doing. Abraham and David both sinned for neither were perfect. Though not perfect, they both were capable of having an obedient faith and justified by that faith. Paul quoted David in Romans 3 and David was an obedient man who sought after God and was righteous.
3) "There is none righteous"
There were men who were righteous as Abraham, David and Abel (Hebrews 11:4) but there were none righteous in the absolute sense. Again, the OT law did require absolute perfect righteousness to be justified by it, yet no Jew could keep it absolutely perfect hence 'none righteous' in the absolute sense required by the OT law. Abraham,David and Abel were justified instead by a faithful obedience. Those under the OT law who had a faithful obedience had all their sins cleansed away by Christ when Christ died and shed His blood (Hebrews 9:15). When Christ shed His blood it flowed backwards to cover the sin of those under the first covenant, those who were obedient. Hence their righteousness came by the shed blood of Christ washing away all their sins making them righteous through Christ due to their faithful obedience.
4)"there is none that seeketh after God"
There were those who did seek after God as Abraham and David. Men do have the free will to "set their heart" to choose to seek the Lord or not (1 Chronicles 22:19; 2 Chronicles 12:14). Men are commanded to seek God (Isaiah 55:6), no sense to command men to seek God if man cannot possibly do so. Paul said of men "That the residue of men might seek after the Lord, and all the Gentiles, upon whom my name is called, saith the Lord, who doeth all these things" and "And hath made of one blood all nations of men for to dwell on all the face of the earth, and hath determined the times before appointed, and the bounds of their habitation; That they should seek the Lord, if haply they might feel after him, and find him, though he be not far from every one of us:" Therefore man does have the ability, but also the responsibility and accountability to seek after God.
Again, Paul was specifically addressing the Jews in Romans 3:10-18 for their not seeking God. Paul was attacking the Jew's hypocrisy. The Jews prided themselves on being students of the law, studying the law daily and having great knowledge of the law, when in reality they did not know the law....
"....of all the sins the Jew considered himself above, it was spiritual ignorance due to a failure to seek God; and yet, right here it was in their own Bible. They neither understood nor sought after God. True, they knew many things; but they had never understood that their entire system was temporary, typical, and comparable to the scaffolding of a building, and due to be torn down when the great antitype was revealed. They had somehow missed the overriding fact that Judaism was not designed to be God's permanent order of things." Coffman Comm.
"There is none that seeketh after God ... What a paradox was this, that the chosen nation who had received the revelation of God and who had studied it so meticulously, were, in all that study, not seeking God at all, due to the lack of any proper motive, and having forgotten the warning of Hosea, "Then shall we know, if we follow on to know the Lord" (Hosea 6:3). Knowing what the scripture says is one thing; following on to know the Lord is another. Since the Jews were not seeking after God, what was the point of all their study? Christ himself pinpointed the trouble: it was this, that they desired the praise of men rather than the praise of God (John 12:43)." Coffman Comm
Jesus Himself called these Jews "blind" (Matthew 23) and said they did not seek things from God (John 5:44) but sought to glorify themselves. Paul in Romans 3:10-18 is giving a scathing rebuke of the Jews..."Notice the accusations against the proud Jewish race (Romans 3:12). Despite their access to the word of God and their devotion to study, God says they "lack understanding" (Romans 3:11). Despite having in their possession the word of God, they didn't use it to 'seek' Him (3:11)" Dunagan Comm.
5) "They are all gone out of the way, they are together become unprofitable; there is none that doeth good, no, not one."
Paul does NOT say they were BORN out of the way or BORN unprofitable but that have gone out of the way, they become unprofitable showing personable culpability not how one is innately born.
"Their throat is an open sepulchre; with their tongues they have used deceit; the poison of asps is under their lips: Whose mouth is full of cursing and bitterness: Their feet are swift to shed blood: Destruction and misery are in their ways:" These words are used to describe the sinful Jew, not referring to new born infants who cannot speak, curse, shed blood, etc.