Let us look first at this verse:
"Who also hath made us able ministers of the new testament (diatheke); not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life" (2 Cor.3:6; KJV).
This is not about the New Covent promised to Israel but instead it is about the Lord Jesus' Last will and Testament. That "will" is synonymous with the gospel. And it is the gospel which gives life.
John Calvin certainly understood that Paul was speaking of the ministry to preach the gospel, writing the following:
"'Not of the letter but of the spirit'...There is, however, no doubt, that by the term 'letter,' he means the Old Testament, as by the term 'spirit' he means the gospel; for, after having called himself a 'minister of the New Testament,' he immediately adds, by way of exposition, that he is a 'minister of the spirit,' and contrasts the letter with the spirit" [emphasis added] (John Calvin, Commentary on Corinthians - Volume 2, Christian Classics Ethereal Library).
In the Jamieson-Fausset-Brown Bible Commentary we read the following remarks on 2 Corinthians 3:6:
"spirit giveth life--The spirit of the Gospel when brought home to the heart by the Holy Spirit, gives new spiritual life to a man (Ro 6:4, 11)" [emphasis added] (Jamieson-Fausset-Brown Bible Commentary, Commentary at 2 Corinthians 3:6).
We are not ministers of Israel's New Covenant but instead we are ministers of the gospel of the grace of God, as witnessed by Paul's words here:
"But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God" (Acts 20:24).
Every time a New
Diatheke is referred to the Body of Christ the reference is to the "gospel" and not the New Covenant promised to Israel.