I believe the Wycliffe translation best represents the original text:
the which not of bloods [which not of bloods], neither of the will of flesh, neither of the will of man, but be born of God. (WYC)
The entire statement taken with the entire Gospel in view, speaks to being born again:
Yet to all who received Him, He gave the power to become sons of God, to those who believed in His name, who were born not of blood, nor of the will of the flesh, nor of the will of man, but of God. (John 1:12-13) [MEV]
Jesus answered him, “Truly, truly I say to you, unless a man is born again, he cannot see the kingdom of God.” Nicodemus said to Him, “How can a man be born when he is old? Can he enter a second time into his mother’s womb and be born?” Jesus answered, “Truly, truly I say to you, unless a man is born of water and the Spirit, he cannot enter the kingdom of God. That which is born of the flesh is flesh, and that which is born of the Spirit is spirit. Do not marvel that I said to you, ‘You must be born again.’ The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit.” (John 3:3-8)
This is the Gospel, which is for all people:
But these are written that you might believe that Jesus is the Christ, the Son of God, and that believing you may have life in His name. (John 20:31)
And this gospel of the kingdom will be preached throughout the world as a testimony to all nations, and then the end will come. (Matthew 24:14)
In the Kingdom of God people are divided into two main groups, those who are Gentile and those who are Jewish. Those born of the will of the flesh are described as from σάρξ (flesh). This is the word used to describe those who are of Jewish descent:
concerning His Son, Jesus Christ our Lord, who was born of the seed of David according to the flesh (σάρκα) (Romans 1:3)
For I could wish that I myself were accursed from Christ for my brothers, my kinsmen by race (σάρκα) (Romans 9:3)
The two primary groups are those born of the will of the flesh, who are Jewish, and those born of the will of man, who are Gentiles.
Those who are of Jewish descent may be divided into two groups, Levites and all others. The Levites inherited no land as their inheritance was service to the LORD in the Tabernacle and Temple. The Levites were chosen in place of the first born:
I Myself have taken the Levites from among the children of Israel instead of all the firstborn that open the womb among the children of Israel. Therefore the Levites will be Mine because all the firstborn are Mine, for on the day that I struck all the firstborn in the land of Egypt I set apart to Me all the firstborn in Israel, both man and beast. They will be Mine. I am the LORD.
In Egypt, a firstborn (singular) was saved by the blood used to mark the door posts to their house. Among the nation then, all the first born (plural) had been saved by the bloods of the lambs killed to mark the door posts. Therefore, the first born (plural) had been saved by bloods. When the LORD exchanged the first born who were saved in Egypt with the Levites, in that sense they would be considered to have been born of bloods.
The four types of people listed in order are:
Levites: born of bloods (first born ---> Levites)
Jewish: born of the will of the flesh
Gentiles: born of the will of man
Children (of God): born of God (Levite or Jewish or Gentile ---> Children of God)
The order and the use of "will" is likely significant. Levites, Jewish, Gentile are listed in inverse order of how the group came to be. Levites were born of bloods yet in another sense they were "chosen" of God, which how children of God come to be.
The three types are Levites, Jewish, and Gentiles. Regardless of what type of human ancestry is used to characterize a person, they have been given the right to become children of God.
- Or "born from above" or "born again...from above"