This question is related to the eternal begetting of the Son NOT his incarnation.
If God's nature includes being without origin how is it his Son, with an origin, is equal to that nature?
Nature doesn't have a personal property such as paternity, filial, and spiration. These hypostatic (personal) property belong to the three divine hypostases not their common essence (homo ousia). This is why Arius rejected homoousia in Nicene council because he can't see how the Son who has an origin from the Father is equal with the Father. This Trinitarian issue still exist from Nicaea (325) until it's solved at Fourth Lateran (1215). To shed a little bit history let us visit the Fourth Lateran Council:
Here, Joachim accused Lombard for inventing a fourth person of Trinity namely the divine essence. Lombard was exonerated and Joachim was condemned. This is how Fourth Lateran explains the Orthodoxy of Lombard in the same canon:
The Son who has a hypostatic origin begotten from the Father receives the same nature that His Father has. This way the Son is distinguishable from the Father hypostatically but not essentially. Just as a human father can be distinguished from a human son hypostatically but not essentially. The different is a human father can only share a portion of his nature to his son this is why a son is like (in Greek homoios) his father. While the Father share the same nature (homo ousios) in its totality not partial to His Son. This is why in term of essence the Son is identical with the Father, homoousia.
Arius can't see how the Begotten Son can be equal with the Unbegotten Father. Lombard developed a clarity between a hypostatic property of Unbegotten belongs only to the Father and His essence which doesn't have any hypostatic property. The divine essence clarified by Lombard is neither unbegotten nor begotten because those are hypostatic properties not essential. This is why the divine essence shared by the three divine persons is anhypostatic it has no personal property. In modern view of Reformed Trinitarianism developed by a Dutch American Cornelius Van Til he revived a view that God is three persons and one person referring to the divine essence as a hypostasis which the Fourth Lateran condemned. We see how Nicene faith, the faith of the Apostles is kept through present time through the Fourth Lateran Council faithfully that the divine essence is not a person.
... his Son, with an origin,....
This is one of the classic misconception of Trinity. Jesus is not begotten in a particular point in time by The Father. As you have mentioned Jesus is eternally begotten. That is, in the very nature of God's existence, Jesus was/is/will-be begotten by the Father for eternity. This is same as how holy spirit was/is/will-be proceeding from the father (and the son). That is being eternally begotten is not an event in the past, rather the state of existence (being) of second person of the Trinity in the Trinity.
John 1:1,14 is the classic proof text of the eternality of the Son.
John 1:14 goes on to say that this is Jesus.
Furthermore the creeds are very clear that he was begotten not made, which precisely is there to address the whole subordinationism debate. Basically, the creeds work out in more precise detail how it is that the Son is not subordinate to the Father, by nature of his eternal begottenness.
To go too much further would really be to stumble into book territory, which is beyond the scope of the site.
You are confusing the Deity Jesus with the man Jesus.
When the Angel Gabriel announced to the Virgin Mary that she would become Pregnant and bear a son, he told her that she would be overcome by the Holy Spirit.
Luk 1:35 KJV
The Holy Ghost himself placed the seed of God into Mary's womb which combined with Mary's seed then grew into the baby Jesus.
Because Jesus had both the seed of man which he inherited through Mary his mother and the seed of God which the Holy Spirit placed in the womb of his mother Mary, Jesus was both man and God.
This is no different than the fact that you are the son of both your mother and your father, and have characteristics of both of them.
Mat 1:18 KJV
Mat 1:20 KJV
That seed from the Holy Ghost is eternal therefore the Deity part of Jesus is also eternal.
In NME culture, the firstborn, especially the firstborn son, was very special indeed. He had privileges and perquisites subsequent children did not. Interestingly, one of the titles for Jesus is
This expression does not mean Jesus was created; it does mean that He bears the honorific title of Firstborn, with all the privileges that accrue from being so named. Moreover, He is the creator of all things (ibid., v.16); He existed before all things and holds all things together by the word of His power (ibid., v.17); He is the head of the church, His body; He is the beginning, the firstborn from the dead; He has the first place in everything (ibid., v.18); and all the fullness of Deity dwells in Him (ibid., v.19).
Perhaps the greatest verse to indicate the equality of Jesus with the Father (besides Jesus' own words, "He who has seen me has seen the Father. The Father and I are one person," John 10:30; 14:19; 17:21) is Colossians 2:9,
This verse does not say that Jesus became Divine when He came to earth as a baby, born of the virgin Mary. He was God the Son in eternity, and He never ceased being God the Son, even during the 33 years of His self-emptying life on earth as the God-Man, Jesus Christ (see Philippians 2:7 ff.). Moreover, He will never cease being God, albeit as God in the flesh-and-bone body of a man.
Within the triune Godhead, there is one shining star, and that is Jesus Christ, the only begotten Son of God. No one preceded Him; no one followed Him. He is sui generis, as the French say. While some Roman Catholics will disagree with me on this point, the eternal Son of God, the second person of the trinity, was neither conceived nor born; He simply is:
In eternity past, the Son was the delight of His Father, and the Father of the Son. In other words, they loved one another, and always will. Moreover, each person in the Trinity seems to have a unique, yet overlapping, role to play. Concerning the Father and the Son, Jesus said,
While your question bypasses the role of the Holy Spirit, He too has been delegated a role; namely, to reveal Jesus to people by removing the blindness from their spiritual eyes and by imparting Jesus' words of life to them, first, when they are first born from above, and second, as they grow in Christ. Christ, in other words, is formed in us by the unseen work of the Holy Spirit of God during the lifelong process of sanctification.
The Trinity is indeed a mind-blowing doctrine, neither easily explained nor easily understood. It is, however, part of the bedrock of the Christian faith. The only acceptable sacrifice for sin had to be both fully God and fully man, the infinitely perfect, spotless, and holy Lamb of God. Anything less would not suffice to put away sin by the sacrifice of Himself (Heb 9:26).
The Son truly has an origin or source of his very existence. He is not dependent of himself for existence because he himself is a son. He depends on his Father for existence.
" The Son is auto-theos because of his nature not because of his sonship."
The Son has an origin of existence (i.e. the Father) but does not have a beginning of existence. Someone/something who has an origin can still be co-eval with its source.
Example: A heavy ball is placed on a cushion, there's a depression caused in the cushion. The moment it was placed on the cushion was the exact moment when the depression was caused. Hence, the cause and the effect are coeval.
In the Bible, the Father begets the Son. The cause of the Son is the Father but yet he is without beginning of existence because the Father himself is without beginning of existence.
The Scriptures itself in John 1:1-3 and 1 John 1:1-3 stated that the Son, the Logos, is "from the beginning( i.e. God)" for God is the " beginning, origin and source of all things" ( 1 Cor. 8:6).