I'm not being dogmatic here because I am not used to explaining things in these terms. To us, living now, the New Covenant is the only covenant that we directly are involved in. And like I said, most of us as Baptists are unfamiliar with that terminology although I don't see anything wrong with it.
As a teaching tool, the covenant of redemption is between the Trinity and so to us is mostly instructive in nature. If you are a strict Calvinist is lays a framework for a deterministic view of salvation. The covenant of works and the covenant of grace are the main overlaying framework of God's dealing with men as you read the whole Bible. I have found that studying the Old Testament in that framework (that an overlaying covenant of grace was in operation all along) seems to bring the Old Testament alive to me as a Christian in 2023. Of course it all culminates in the work of Christ but NCT to me anyway seems to devalue the Old Testament stories as having any relevance other than background and foundation. Like I said, I like Piper and know some folks who I think went to Clear Creek when this was being taught (NCT) and I'll ask them next time I see them but I know they are solid for sure. (I could be completely wrong on the definition of the covenant of redemption because all I know of it came from that great theologian called "the internet".
I think personal experiences comes into play. I am a traditional Baptist so the New Covenant is something I have always assumed to be the greater covenant (even before I read that in Scripture).
I have found the opposite to be true in terms of CT and NCT.
NCT and CT both treat the Old Testament seriously. But they do so differently. Trading Scripture as a narrative - Adam's sin, God's covenant with Noah, then Abraham, then with Israel (the "Old Covenant'), then with David (kingship) and those progressing to the Promise revealed in the New Covenant makes the Old Testament more alive to me than imposing a system into Scripture could.
Now, if you were to say that Covenant Theology reinforces ideas specific to Reformed Theology then I would agree. But I believe that detracts from Scripture itself.
I think you misunderstood some of NCT. It does not devalue the Old Testament at all (NCT churches I've attended seen to preach and teach more from the Old Testament than the New Testament....I'm not sure why).
And a major point of NCT is that the Old Covenant is not the foundation for the New Testament but it's the other way around. The Old Testament was a shadow of what was to come - it looked to the Promise fulfilled with the New Covenant.
Have you ever considered the possibility that the three primary "covenants" in CT could be wrong? Covenant Theology is a relatively new idea (in terms of Christianity) and those three "covenants" are not actually in the Bible. If men made a mistake then you are laying a false framework over Scripture which could mean you are missing a lot of the depth that is in the biblical narrative.