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I really have not thought about whether or not the Old Testament covenants formed the foundation on which church discipline was established, but there has to be a certain amount of truth to this since the New Covenant is ultimately the fulfillment (at least partially with kingdom promises still to be fulfilled) of those covenants as they anticipated the reign and rule of Messiah while also providing a tutor for Israel until Christ. I’ll share a few thoughts, but this is just off the top of my head. This is certainly not my area of expertise.
Both freedom and law or discipline have their roots in the Old Testament covenants and their promise of liberty and justice through righteous laws and eventually through the Messiah, the righteous judge. This means there must be some discipline or governmental controls to promote true liberty which, as I understand it, means not the freedom to do as one pleases, but the ability to do what one ought. This is where the New Testament comes so strongly into focus with grace, the enabling capacity to produce the righteousness of the law in the power of the Spirit (Rom. 6:14; 8:1f). However, because even regenerate man is sinful, there is the need of some form of discipline to protect the church, as with Israel, that it might be kept pure or remove the leaven. Of course, we have failed miserably in this either by neglecting church discipline or by its misuse through legalism (setting up human taboos) and by dictatorial boards that often try to run the lives of its members far beyond the directives of Scripture.
While the New Testament still anticipates the return and righteous rule of Messiah in the literal kingdom on earth, He rules now over the affairs of the world from heaven by His sovereignty allowing the world to move toward its natural end according to God’s plan and wisdom. But the Lord also seeks to rule in the lives of His people through His Word, the ministry of the Spirit, and through the body of Christ, the church and its government (pastor/elders and flock). Being dispensational, I firmly believe that the church is distinct from Israel, a mystery in the Old Testament and a unique undertaking of God in this present economy, but this does not do away with the principles of righteousness, liberty, and justice as set forth in the Old Testament covenants, particularly the Law and the Ten Commandments. That is our foundation, but while we are not under the Law as a code, its character is to be produced and manifested by the Spirit, for where the Spirit is, there is liberty (the ability or power, by grace, to do as we ought without misusing our liberty) (Rom. 8:1f; 2 Cor. 3:17; 1 Cor. 8:9; Gal. 5:1-13).
So surely, there is a carry over here. There are several articles in Bib Sac on the nature of the Law and its impact on us today and these might be helpful. There are also articles critiquing Reconstructionism that might help as well.