That's correct. The law was always supposed to point people to Christ. Fallen man's utter inability to keep the law was supposed to cause Him to fall on His face before God and cry out for mercy. But Israel was under the old covenant and God was dealing with them as a nation. And individually, and corporately, they failed miserably to keep the covenant. So God had no choice but to pour out His wrath upon themJesus answered, “I am the way and the truth and the life. No one comes to the Father except through Me. John 14:6. Jesus didn't teach Old Covenant- He taught how to enter the narrow path back to Him- He died for our sins and lived as our example to follow 1 John 2:6 He is the Way- there is no other door.
Paul was a servant of Christ and never taught another gospel than what Christ taught. Jesus commissioned the apostles to observe everything He commanded. Mat 28:18-20 Paul never taught anything different, which is why we have a warning on Paul’s teachings 2 Peter 3:16
The new covenant isn't between God and men. The new covenant is between God the Father and God the Son. That's why it is an everlasting covenant that cannot fail. So new covenant saints are not under the law, but are kept by the grace and mercy of God. The only thing required to take part in this covenant is belief. If one believes, they are placed in Christ. And as He will never fail, those in Him will never die. Easy peasy.