Let's be clear, the Jews never believed that the Law would save them. Paul never believed that obeying the law would save him. The Jews always believed that the Messiah would save them. They believed that if the entire nation would keep the entire law for one day, then the Messiah would come. Then he would save them from their enemies.
So then, the burning question in the minds of the Jews, at that time, was whether or not one stood condemned before the law or not. Was Paul, the individual, blameless before the law or not? He claims to have been blameless before the law. One day, according to him, he found that he wasn't. He describes this as "I was once alive but I died." Not only did he find that he was condemned before the law, he found that it was impossible for him to avoid coveting. If HE could never live a day without sin, then surely the rest of his nation wouldn't be able to do that either. His entire nation stood condemned and there was NO hope that God would send a messiah to save them. (In their minds, they didn't need to be saved from sin; they needed to be saved from their enemies.)
After explaining the hopelessness of the situation, Paul starts chapter 8 with the announcement that those in Christ Jesus are free of condemnation. "For the law of the Spirit of life in Christ Jesus has set you free from the law of sin and of death." Paul discovered that it wasn't the law that brings life; it is the Spirit that brings life. And once an individual is seeking life by the Spirit, he no longer needs to seek life by a Law that is only able to condemn one to die. Obeying the law could never absolve anyone of condemnation. He proved that in chapter 7.
I don't disagree with you. But I wasn't answering that question. Paul says that the law itself was weak. And I was coming to terms with THAT idea. I understand that WE are weak, of course. But Paul was arguing that the Law itself had a weakness.