One of the clearest institutional types that prefigure Christ is the is institutional priesthood. One can see the escalation from lesser to greater quite clear in the typology (and biblical theology) of priesthood that Hebrews presents.
As Hebrews 5:1 indicates, a high priest “is appointed to act on behalf of men in relation to God.” In other words, they are mediators between the one holy God and sinful man. No man chooses for himself to become a priest, but is called (Heb 5:4). Likewise, “Christ did not exalt himself to be made a high priest, but was appointed” (5:5). However, a great distinction can be made between the levitical high priests and Jesus Christ.
The high priest “is beset with weakness. Because of this he is obligated to offer sacrifice for his own sins just as he does for those of the people” (5:2–3). However, Jesus “in every respect has been tempted as we are, yet without sin” (4:15). As this perfect one, Jesus doesn’t need to offer up a sacrifice for himself (7:27). And so as a son who is perfectly obedient, Jesus “became the source of eternal salvation to all who obey him, being designated by God a high priest after the order of Melchizedek” (5:9–10).
The author of Hebrews goes on saying, “Now if perfection had been attainable through the Levitical priesthood (for under it the people received the law), what further need would there have been for another priest to arise after the order of Melchizedek, rather than one named after the order of Aaron” (7:11)? The author of Hebrews, here, touches on the Levitical order’s insufficiency. The author goes on to speak of this priesthood’s inadequacy: “The former priests were many in number, because they were prevented by death from continuing in office” (7:23).
On the other hand, Jesus’s priesthood has one enduring priest—Jesus: “he holds his priesthood permanently, because he continues forever” (7:24). As the resurrected Son of God who sits at the right hand of the throne of God (1:3), Jesus will never die again (cf. Rom 6:9). And so the author of Hebrews says, “he continues forever” and therefore his priesthood will never end. As one who endures forever, the resulting priesthood is both astonishing and assuring: “he is able to save to the uttermost those who draw near to God through him, since he always lives to make intercession for them” (Heb 7:25).
The former, Levitical priesthood served as a type of (pointer to) Christ. It paved a pattern that revealed we needed a representative—a mediator. But unlike the priests who die and need to be replaced, Jesus never dies, and thus he is able to make eternal intercession for his people as the one Mediator between God and man (1 Tim 2:5). Jesus as priest did not offer up the blood of goats and bulls, which really does not take away sins (Heb 10:4), but he offered up his own body as the perfect and once-for-all sacrifice that truly is effective. The priest “stands daily at his service, offering repeatedly the same sacrifices, which can never take away sins. But when Christ had offered for all time a single sacrifice for sins, he sat down at the right hand of God, waiting from that time until his enemies should be made a footstool for his feet. For by a single offering he has perfected for all time those who are being sanctified” (10:11–14).
Truly the priesthood in the OT was inadequate, and Jesus fulfilled it as the true and lasting High Priest.