This piece by Jared Longshore is an archetypal example of paedobaptist argumentation in important aspects.
What I mean by that is as follows. He begins by defining some theological abstractions, and their characteristics (N.B. by this I am not implying that theological abstractions are bad things, or to be avoided in general). Then, he moves from those abstractions to discuss the Biblical evidence in their light. And along the way, he asserts that the abstractions already require, before the New Testament begins, what conclusions can validly be drawn from the New Testament data.
This is the wrong way to approach the question he's addressing. Instead, you should begin with the New Testament's various direct and deliberate explanations of how the New Covenant works, and how it relates to other covenants. From there, you should then work back towards your higher-level/deep abstractions about how God's ultimate covenant of salvation works.
If you fail to do this the right way, then what's going to happen is that your a priori abstractions are going to control the interpretation of the data, instead of being built from the data.
The covenantal paedobaptist may assert that he if he were to do things the way I say, he'd still get the same result, but this raises the question: why doesn't he? Or if I chose to be ruder: actually I don't believe him. If paedobaptist conclusions could be built by starting with the Bible's clear and deliberate explanations of how the New Covenant works, and how it relates to what came before (and what comes "behind"), then paedobaptists wouldn't always choose to carry out their argumentation another way. (And note that the above link is the opening statement of a debate about baptism; it is a basic of debate that you should present your strongest case in relation to your opponent's argument, so this would have been exactly when to do so). Instead, by laying down arguments which then control, a priori, how the New Testement should be read, the Bible is muzzled, and the conclusion is pre-determined by the higher-level abstractions which come in prior to the detailed exegesis, and control its boundaries.
That's how you end up - even in a debate when presenting your strongest arguments - with statements from the paedobaptist side like this one from the above, "thus the burden is on those who would change God’s covenantal pattern in the new covenant". Baptists don't propose to "change" anything. That's a loaded statement. Since the New Covenant contains the full and final revelation of God's purposes, and illuminates, explains and clarifies what was previously in the shadows, it guides the interpretation of what came before and our doctrine of precisely how they relate. Understanding the New Covenant is required first in order to reveal to us how the covenants relate.
This loaded statement also assumes what needs to be proved in multiple ways. How do you know that "God's covenantal pattern" must mean that when the New Covenant comes, the category of "children of Abraham" is going to include the one-generational offspring of Gentile believers? There's a lot of issues of continuity and discontinuity of different kinds that have to be worked through before you can get there. Happily the New Testament has more than enough to get us to the correct conclusion. But you certainly can't conclude "one-generational offspring of Gentile believers are 'in the covenant', and also all receive any covenant sign" without detailed New Testament exegesis, so it can't be asserted a priori. Any argument which asserts it a priori must be, ironically, a priori wrong. And indeed, such an assertion proves too much: if there is indeed (as I believe there is) a final covenant involving the Triune Godhead and the elect, then such a strong assertion would have to result in the conclusion that these one-generational offspring are not only entitled to any covenant sign, but also actually and definitely and always finally saved. Otherwise, we're supposed to conclude that all the "administrations" of the covenant are critically different (at the point which the writer's whole argument is to insist is key, the point which the whole debate is about) from the final covenant itself. That is to say: his argument proves too much; he's ultimately obliged by it ultimately to argue (something which we know is false), that all one-generational offspring of believers are certainly finally saved; otherwise we have precisely the sort of change which he is arguing is already known to be impossible by the time that the Old Testament closes.
"Abraham's physical descendants through the lines of Isaac and then Jacob were all in covenant with God, and their male offspring were thereby required to be circumcised; therefore all the next-generational offspring of Christian believers are also members of the New Covenant and thus should be baptised" cannot be proved by a priori argumentation about God's "covenant of grace" and assertions about how "administrations" of that covenant "must" work. It has to be proved by demonstrating, by good and necessary inference, that this is how the apostles actually explained that the New Covenant operates upon the relevant points of detail. They didn't do this, and their actual explanations point in a different direction.
