Romans 10:3
"For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God."
There is a difference between doing one's OWN righteousness that does not justify and doing GOD'S righteousness that does justify. Romans 10:1-2 those Jews were lost for they were doing their OWN righteousness and not doing GOD'S righteousness.
"not having mine own righteousness, which is of the law"
Earlier in Paul's life before converting to Christianity when he was Saul, he as other Jews tried to find righteousness in the eyes of God by the OT law by working to keep it perfectly...their own righteousness which is of the law.
But now Paul knows trying to find righteousness by perfect law keeping is vain, (Romans 7:10), in that no Jew could keep the law perfectly (Romans 3:23) therefore no one can be justified in the sight of God by the law (Galatians 3:11)..."But that no man is justified by the law in the sight of God, it is evident: for, The just shall live by faith." - (Habakkuk 2:4). Hence Paul DID find justification by faith. Meaning God, under both the OT and NT, was looking for justification by an obedient faith, not by perfect, flawless law keeping. The OT law did not even require faith (Galatians 3:12).
Therefore righteousness comes through faith in Christ and not by one's own righteousness in perfect, flawless law keeping. So a right standing before God is by an obedient faith and not faith only (James 2:24) not by works of merit (Ephesians 2:9; Titus 3:5). not by flawlessly keeping the law of Moses (Galatians 3:11-12).
A right standing with God is by "submitting to the righteousness of God" (Romans 10:3) meaning "the faith" of Philippians 3:9 must include repentance (Luke 13:3) confession (Matthew 10:32-33) and baptism (Mark 16:16). "the righteousness which is of God by faith" - righteousness comes from God not by works of merit, not earned, not from flawless law keeping but by an obedient faith.
Philippians 3:9 "And be found in him"
"As Boice said, "The are two kinds of righteousness,"[19] that which comes of men, and that which is achieved by God. The righteousness which saves was not achieved by people, but by Christ; therefore, it is called here the "righteousness of Christ," or the "righteousness of God through the faith of Christ." One who wishes to be saved must become a participant in the righteousness achieved through the faith and perfect obedience of Christ. For five hundred years, the monstrous heresy has prevailed that people achieve that perfect righteousness merely through believing subjectively in Christ, Paul revealed how truly people become sharers in Christ's righteousness. They renounce self, deny themselves, believe in Christ and obey the gospel by being baptized into Christ, thus becoming Christ, in the sense of being "in him" and identified with Christ. The righteousness that saves is not theirs but Christ's; and even in the case of Christ's righteousness, it was not achieved by faith only but by faith and our Saviour's perfect obedience. Thus every man who will be saved shall not be saved as 'Joe Doakes', but as Jesus Christ."
Coffman Comm.
(19) James Montgomery Boice, op. cit., p. 200.
"but that which is through the faith of Christ"
If it were not for the perfect faith, perfect obedience of Christ, salvation could not be had. But because Christ lived a perfect righteous life, man can have salvation through Christ's perfect righteousness. Therefore to be saved through the perfect righteous faith of Christ one must be "in Him" thereby be identified with Christ and HIS perfect righteous faith.