Thank you for taking the time to consider this. It is important that we fully grasp what Paul is teaching about the new and old covenants. So much of his message in the New Testament is bound up with the law and grace. He was speaking to a church that was very legalistic. I believe it had a lot to do with his choice of words and examples.
I think this is why he does not bring into account the everlasting covenant. The Jews were set on what God had given them at Sinai. They held the "law" in great esteem and then killed the author of that law. Jesus came into this world to correct their error in regards to the law. They thought He was doing away with it, but not so. He was only sweeping away the mess they had piled on top of it. Human laws that brought reproach upon His holy law and made the keeping of the law burdensome.
Elsewhere Paul tells us that the law is holy, just, and good. And that it was glorious. But, it is only the "schoolmaster" that leads to Christ. Christ, is the Glorious One whose character is seen in the law. It appears to me that Paul contrasts the old and new covenants in this light. The law given at Sinai was glorious, but rather than have the law written on tables of stone, the new covenant promised to write it on the fleshly tables of the heart. A much better covenant established upon better promises.
It is true that the old covenant shadowed forth Christ, but after the cross, there was no more human priest standing between man and God. Now man could bring his sacrifice directly to God through our living High Priest who was mediating in the heavenly sanctuary. In the everlasting covenant, the very same promises existed as in the new covenant, that God would write His law in the heart, He would give us enmity towards sin we did not possess without Jesus in the heart. The old covenant was a bridge to the everlasting covenant. Through the sanctuary and its services, a people who were little more than brute beasts coming out of Egypt were to learn of the plan of salvation and their need of a Saviour. They had lost sight of it all and they needed to learn of their great need. They thought they could keep the law on their own. The old covenant was to teach them they needed a Saviour. The law, both moral and ceremonial was to do its work as a schoolmaster. This was the "ministration of death". Then the Israelite as well as the gentile could claim the promises of the everlasting covenant which is the "ministration of life", life in Christ Jesus.
Why is this important today? Well, reading the writings of Paul is hard. But, when understood, we receive a great blessing. He wrote much of the New Testament. Besides, think on poor Peter. He loved Jesus, but did not know his own heart. He did not know of his dependence upon Christ to do any good thing. Self assured he was. Then I come down to our day. Do we see a similar situation in the church, in God's church? We do. It is in a Laodicean condition where most believe they are rich and increased with goods, but know not that they are miserable, wretched, poor, blind, and naked. The very same self assuredness that plagued poor Peter. We all need to come to the point in life where we proclaim with Saul, "O wretched man that I am, WHO will deliver me from the body of this death?" When the law has done its work, then we are ready to ask the all important question "WHO" will help me?
The law has not been done away with. It was the ministration of death that was done away with. The old covenant was no longer needed since Christ has come and suffered and died. Away with the whole system of priests and laws that shadowed forth the Saviour. He had come and had left not just a teaching and an example, but He has given His Spirit to live in man. The Spirit had always been there and all who were saved before the cross were saved in the very same manner we are today, by the everlasting covenant, by the blood of Christ, by dying to self and living to God through the abiding presence of Jesus Christ in the heart. Ezekiel made it perfectly clear, A new heart will I give you, a new Spirit will I put within you! I will cause you walk in my statutes. This is the better covenant established upon better promises than those made at Sinai, obey and live.
We have much to learn and I am only scratching at the surface of the righteousness of Christ and all that He means to us. It is present truth! It is the third angels message in verity! The loveliness of Jesus!! The law is glorious, but Christ is much more glorious!! He is the reality!