Original Sin and Titus

It is common for Churches to teach it was impossible for man to do good without Christ.

The Wesleyan church for instance:

We believe that humanity’s creation in the image of God included ability to choose between right and wrong. Thus individuals were made morally responsible for their choices. But since the fall of Adam, people are unable in their own strength to do the right. This is due to original sin, which is not simply the following of Adam’s example, but rather the corruption of the nature of each mortal, and is reproduced naturally in Adam’s descendants. Because of it, humans are very far gone from original righteousness, and by nature are continually inclined to evil. They cannot of themselves even call upon God or exercise faith for salvation. But through Jesus Christ the prevenient grace of God makes possible what humans in self effort cannot do. It is bestowed freely upon all, enabling all who will to turn and be saved.

Paul's letter to Titus acknowledges man could work works of righteousness before becoming a Christian. It is just that works of righteousness had no power to remove sin, which we also have.

Man could both sin and work righteousness before Christ came into the world..

Tit 3:5 Not by works of righteousness which we have done, but according to his mercy he saved us, by the washing of regeneration, and renewing of the Holy Ghost;

Paul acknowledged man had done works of righteousness.

Psa 15:1 A Psalm of David. LORD, who shall abide in thy tabernacle? who shall dwell in thy holy hill?

Psa 15:2 He that walketh uprightly, and worketh righteousness, and speaketh the truth in his heart.

Psalms 7 declares righteousness is in man just as some verses say sin is in man.

Psa 7:8 The LORD shall judge the people: judge me, O LORD, according to my righteousness, and according to mine integrity that is in me.

Again, works of righteousness didn't save man because they did not remove the sin we also had. Removal of sin is according to God's mercy by Christ's death and baptism in the likeness of Christ's death.

Working righteousness did not imply perfect in righteousness, it did not imply sinlessness, it just meant you had done things right in some instances. A person who sought to follow God.