Now a question may arise in your mind, Why did God bother to go through the whole "first level" or typological fulfillment in the first place? If the "real" narrative is the Adam narrative of creation, fall, redemption, consummation, why Israel? I believe the reason is pedagogical. The Adam narrative happened too long ago and with so few witnesses. So God re-enacted the Adam narrative on the grand stage of world history in order to set the context for the incarnation, death, and resurrection of Christ.
Meredith Kline put it this way:
A variety of purposes can be discovered to explain the insertion of the old covenant order and its typal kingdom into the course of redemptive history. Of central importance was the creation of the proper historical setting for the advent of the Son of God and his earthly mission (cf. Rom 9:5). In accordance with the terms of his covenant of works with the Father he was to come as the second Adam in order to undergo a representative probation and by his obedient and triumphant accomplishment thereof to establish the legal ground for God’s covenanted bestowal of the eternal kingdom of salvation on his people. It was therefore expedient, if not necessary, that Christ appear within a covenant order which, like the covenant with the first Adam, was governed by the works principle (cf. Gal 4:4). The typal kingdom of the old covenant was precisely that. Within the limitations of the fallen world and with modifications peculiar to the redemptive process, the old theocratic kingdom was a reproduction of the original covenantal order. Israel as the theocratic nation was mankind stationed once again in a paradise-sanctuary, under probation in a covenant of works. In the context of that situation, the Incarnation event was legible; apart from it the meaning of the appearing and ministry of the Son of Man would hardly have been perspicuous. Because of the congruence between Jesus’ particular historical identity as the true Israel, born under the law, and his universally relevant role as the second Adam, the significance of his mission as the accomplishing of a probationary assignment in a works covenant in behalf of the elect of all ages was lucidly expressed and readily readable.
Much more than the works-probation aspect of Jesus’ task was included in the revelatory design of the typal kingdom. It prepared a public context in world history in which the meaning of Jesus’ mission as a whole might be communicated effectively.
[Meredith G. Kline, Kingdom Prologue (Eugene, Oregon: Wipf and Stock, 2006), 352-3.]
Note the statement highlighted in bold: "the congruence between Jesus' particular historical identity as the true Israel under the law, and his universally relevant role as the second Adam." That is the heart of the matter.
I would argue that the role of the Davidic king is what most clearly points backward to Adam and forward to Christ as the second Adam. It was the king who was responsible to lead the people of God in faithful obedience to the Law. As the king goes, so go the people. If he is obedient, the people are generally faithful and their tenure in the land continues. If he is an idolater, the nation as a whole is filled with idolaters and is (eventually) cut off from the land.
God said to King Solomon after he finished building the temple: "If you turn aside from following me, you or your children, and do not keep my commandments and my statutes that I have set before you, but go and serve other gods and worship them, then I will cut off Israel from the land that I have given them, and the house that I have consecrated for my name I will cast out of my sight" (1 Kings 9:6-7 ESV).
The king's disobedience or obedience, then, in a typological and imperfect way, is what determines whether God's people continue to dwell in God's holy land. In the same way, Christ's obedience (as God's anointed King) is the meritorious ground that secures the eternal inheritance. Therefore, to call Jesus "the Messiah" is to call him "the second Adam" using the idiom of the typal kingdom.
The cross fits in as follows. The obedience of the Messiah or second Adam included a mission of not only fulfilling the precepts of the Law but of enduring the Law's curse on behalf of those who had violated the Law. This of course would not have been necessary for the pre-fall Adam nor was it envisioned as part of the Davidic king's ordinary role. But now that the covenant of works (whether in its Adamic or Mosaic form) has been broken, the only way the blessings of the covenant can be enjoyed by sinners is if they are judged under the penal curse of the covenant and emerge on the other side through union with a representative Substitute in his death and resurrection. The kingly office of Christ is then conjoined with the priestly.