No, the context of Jesus' discussion escapes you. The rich man was still under the old covenant which required the following of the law. After Jesus' sacrifice and resurrection the new covenant comes into effect and salvation is by grace and not works - read your bible
Ephesians 2:8-9) - "For by grace you have been saved through faith; and that not of yourselves, it is the gift of God; not as a result of works, that no one should boast."
Rom. 3:20, 28) - "because by the works of the Law no flesh will be justified in His sight; for through the Law comes the knowledge of sin...For we maintain that a man is justified by faith apart from works of the Law." (Galatians 2:16) - "nevertheless knowing that a man is not justified by the works of the Law but through faith in Christ Jesus, even we have believed in Christ Jesus, that we may be justified by faith in Christ, and not by the works of the Law; since by the works of the Law shall no flesh be justified."
At which point does the New Testament become the New Testament? After Jesus’ resurrection? So are all Jesus’ comments Old Testament until that point?
You’re really stretching it with this novel approach. No one is saved by their own works. No one is justified of themselves, but only by the salvation and grace offered by Jesus Christ. That said, God does expect work of us. God does expect us to keep His commandments. They’ve never been withdrawn. God will judge us by our works.
You must recognize that the verses you quote are not a rejection of works, but the recognition that nothing starts or progresses in the life of a believer without first faith in Jesus Christ.
That is what the Bible teaches.