This is somewhat vexed and controversial but here are the facts:
Israelite Covenant
The original Israelite covenant expressed in Ex 19 - 23 envisioned no formal priesthood, except for Aaron's high-priesthood. The pre-Sinai practice of the head of each family was a de-facto priest was to have continued. This is confirmed by the opening statements in the Israelite covenant in Ex 19:5, 6 as quoted by the OP.
However, something happened, just what is debatable, but the most likely candidate is the golden calf incident while, of all times, Moses was in the mountain receiving further instructions. This lead to a formal change in priesthood as described below.
Levitical Covenant
The Levitical Covenant is formally stated in Lev 1-9, 16, 21-27 , Num 3, 4, 8, 18, 25:10-13, Deut 33:8-11, Neh 13:29, Mal 2:4-8. This is an eternal covenant (Num 25:12, 13, Ps 106:30) of salt, Num 18:19.
The Levitical covenant is stated in Num 3:11-13, 25:12 where God takes the Levites (especially the line of Phineas as High Priest) instead of the all the firstborn of each family, thus changing the (informal) priesthood from the firstborn of each family to the (formal) priesthood of tribe of Levi. Several Bible writers refer to this Levitical Covenant including:
- Neh 13:29 – “the covenant of the priesthood and the Levites”
- Jer 33:18, 21 – “covenant with the Levitical priests.”
- Mal 2:4-8 – “the covenant with Levi”
- Num 25:10-13 – the eternal covenant of priesthood was also a covenant of peace and a covenant of “salt” (Num 18:19), ie, very solemn and eternal.
- Isa 54:10 & Eze 34:25 also describes the covenant of peace with the Levites
From other Bible passages we can see what the Levitical covenant was.
- The purpose of the Levitical covenant was to teach and inculcate the plan of salvation (Deut 33:9, 10, Heb 9:8, 9, 11-14, 10:1-3, Col 2:16, 17). That is, it symbolically taught about salvation by grace through the coming Messiah via His substitutionary death in the place of the sinner. Thus, the sacrifices, Levites, priests and High Priest became a type of Jesus in various ways.
- A promise by God to set them apart, ie, make them “holy” (Num 3:12, 13), to be a substitute for the first born in Israel, and to have the primary responsibility for caring for the sanctuary, its ritual services and sacrifices, maintaining and transporting its equipment. Ex 32:25-29. In this way, they became the priests of Israel. All these regulations were contained in Lev 1-9, 16, 21-27.
- Thus, the Levites enjoyed a “blessed” (and privileged) status. Ex 32:29, Mal 2:5.
- The book of Leviticus (and Num 1 & 18) sets out the responsibilities of the Levites in operating the ceremonial rites and sanctuary services and caring for the sanctuary equipment. See also Num 1:53, 18:2, Deut 10:8, 31:9, 25, Josh 3:3, 2 Sam 15:24, 1 Chron 15:26.
- The token/sign (Heb: “oth”) of the Levitical covenant appears to have been the unleavened bread at the annual festival (Ex 13:6, 9, 16, Lev 24:8).
The Israelite Covenant never mentions a formal priesthood; thus the Israelite covenant was distinct from the Levitical covenant. In the NT, this Levitical Covenant with all its ceremonies was often abbreviated to “the Law of Moses”, or just, “circumcision” and are clearly understood as distinct from the Old Israelite covenant laws, 1 Cor 7:19, Eph 2:15, Acts 15.
The Transition
The formal change from family-firstborn-priests to Levitical priesthood was done using the mechanism of the redemption money as described (in detail) in Num 3:40-51 and Num 18:14-16.
The New Covenant
When Christ came to fulfill, that is be the anti-type of the Levitical priesthood and high priest of the new covenant, the Levitical priesthood became obsolete. However, the purpose of the Old Covenant is exactly the same as the New Covenant: Specifically, God said of the Christian community, “But you are a chosen people, a royal priesthood, a holy nation, a people for God’s own possession, to proclaim the virtues of Him who called you out of darkness into His marvelous light. Once you were not a people, but now you are the people of God; once you had not received mercy, but now you have received mercy.” 1 Peter 2:9, 10 (Compare Ex 19:5, 6, Hos 1:9, 4:6).
Jesus is the high priest of the New Covenant Heb 4:14-16, 5:10, 7:23-28, 8:1, 2, 9:1-28, 10:1-18. Thus, Jesus inherits and was the fulfillment of the Levitical Covenant (See Mal 3:1). Jesus did this in several ways as well such as:
- Jesus was the fulfillment of what the sanctuary/temple typified, John 2:19-21, Heb 9:1-28, 10:1-18
- Jesus represented the foundation of the temple as well, 1 Peter 2:4-8 (Compare Isa 28:16, Ps 118:22)
- Jesus’ body was represented by the curtain in the temple, Heb 10:20.
- Jesus was the bread of life, John 6:35, 41, 48 (compare Ex 25:23-30, Lev 24:8).
- Jesus was the light of life, John 8:12, 9:5 (compare the lampstand Ex 25:31-39, Lev 24:3, 4, Isa 53:11, Ps 56:13, etc)
- Jesus provides the water of life, John 4:13, 14 (Compare the laver Ex 30:17-21. See also 1 Cor 6:11)
- Jesus is the promised seed of the woman Gal 3:16 (compare Gen 3:15, and the Abrahamic Covenant)
- Jesus was the Passover Lamb and thus the promised Messiah, John 1:29, 1 Cor 5:7, 1 Peter 1:19 (compare Ex 12:1-14).
- Jesus is the High Priest of the New Covenant in fulfillment of the Levitical covenant, Heb 4:14-16, 5:10, 7:23-28, because He was “pure, blameless, set apart” exactly as the Levites were. See also Heb 9:15, 12:24.
- Jesus provided the blood of the new covenant of which the communion ceremony was to be a memorial, Matt 26:28, Mark 14:24, Luke 22:20, 1 Cor 11:25, Heb 13:20, 1 Peter 1:19 (compare Ex 24:5, 8).
- Jesus was the anti-type of the “red heifer” (Num 19:1-10) and its associated ashes that “purify our consciences from works of death”, Heb 9:13, 14.
