I completely agree that the righteous are to live by faith and that becoming righteous through obeying God's instructions is impossible. However, living by faith is about demonstrating that you trust God by living in obedience to His instructions, so it does not refer to living in some other manner that is not in obedience to His instructions. In Hebrews 11, we have a number of examples of people who demonstrated their faith by obeying God's instructions. In Deuteronomy 6:20-25, being careful to obey God's instructions was about demonstrating their faith in God to defeat Pharaoh, demonstrating their faith in God to bring them out of Egypt, demonstrating their faith in God to bring them to the land that He swore to give to their fathers, about demonstrating their faith in God that His instructions are for their own good, and about demonstrating their faith in God to preserve them, so righteousness has never come by the law, but has always come by faith. In Deuteronomy 10:13, God again said that what He command was for our own good, so if you believe God and trust that to be true, then living by faith is demonstrated by living in obedience to His commands.
So it is not obedience to the law that causes us to become righteous, but rather it is our faith in God that does that and our faith that also leads us to become obedient to the law. When God declares us to be righteous by grace through faith, he is declaring us to be someone whose vocation is to train or practice in righteousness by grace through faith (1 John 3:10, 2 Timothy 3:16-17), which again refers to the training that God's grace brings in Titus 2:11-14. God has imputed His righteousness to us so that we will do what He has revealed to be righteous and He saved us by faith through grace from doing what He has revealed in His law to be sin so that we will be free to do what he has revealed to be good works (Ephesians 2:8-10). We have been set free from slavery to what God has revealed to be sin in order to be free to become slaves of obedience leading to righteousness, and slaves of righteousness leading to sanctification (Romans 6:16-19). Our sanctification is about being made to be like Messiah in also doing what God has revealed in His law to be holy, righteous, and good, like he did. When He who began a good work in us is faithful to complete it on the day of Christ Jesus, we will be made live in complete obedience to God's law (Philippians 1:6). The problem with the Old Covenant was not with God's law, but with the hardness of our hearts, so God's plan was to make a New Covenant where he would take away our hard heart, give us hearts of flesh, write His law on our hearts so that we will obey it, and send His Spirit to cause us to obey His law (Jeremiah 31:33, Ezekiel 36:26-27). The law requires obedience and Jesus gave himself so that we might be free to obey the law and thereby meet its righteous requirement (Romans 8:3-4). God is purifying a people for Himself who are zealous for what he has revealed to be good works. God's plan was not to lower His righteous standard, but to make us into people who could meet it by grace through faith.