This question is asked with regard to "Trinitarian orthodoxy", which is where a bit of history regarding the early centuries of the Church reveals a lot as to the matter of beliefs about the two natures of Christ - his deity, and his manhood. The question of when this might have happened is related to this hypostatic union.
Orthodox teaching is that the Trinity means one substance and three persons; the hypostases (persons) of the Trinity share a common ousia (nature). Nestorius (b. after 381) stood against the Arian heresy and supported his chaplain Anastasius who objected to the popular designation of Mary as 'bearer' of God. He taught that the two phyesis (natures) of Christ have different proposa (persons) attached. For Nestorius, the union that makes up Christ has the divine one as eternal and omnipotent, while the human one is mortal and weak. Such 'union of natures' cannot be as strong as that between the persons of the Godhead. He did not want the humanity of Jesus to be swallowed up in the divinity or truncated or in any way made less than like ours. In spite of his attempts to explain how a conjunction of two persons could count as one person (proposon), his Christ turned out to be two individuals and not one. The Son of God did not truly experience human existence "in the flesh" but only through association with the man. Nestorius's Christology was little more than warmed-over and dressed-up adoptionism, as Cyril of Alexandria pointed out (412-44). This is where delving into the teaching of earlier adoptionism will show this clash with orthodox trinitarian teaching. Cyril's unique contribution to Christology is the doctrine of the hypostatic union, in its basic outlines.
(Source: The Story of Christian Theology by Roger E. Olson, pp 217-219)
This gives some sketchy background to the Athanasian Creed of the late 5th century. Orthodox teaching with regard to the duality of the nature possessed uniquely by Jesus of Nazareth, is that these two natures cannot 'merge' or 'mingle'. They are two different things that unite only in the Person of Jesus Christ. Christ's divine and human natures do nor merge or mingle. Both natures meet and unite only in the Person of Christ.
Therefore, the question as to when this happened is linked to why Mary had to be a virgin. The 'conception' is the joining of that eternal
begetting with that which is 'holy', by the Holy Spirit (Matthew 1:20). The Word became flesh at that point, for it was Mary who provided human nature, the One conceived in her womb being fully human, sinless, yet in no way devoid of his eternally existing divine nature. Those two natures met and united at the start of conception, by the power of the Holy Spirit. Those who deny the pre-human existence of this Holy One in Mary's womb have no choice but to alight on some external event that they can claim as a point in time when a human nature became infused with 'something more' from above, and the events at Jordan's banks suit their ideas.
Some claim this only happened at the mature Jesus' water baptism, but you will find that many who claim this do not believe the orthodox Trinity doctrine. Many claim that this Jesus was no more than a sinless, perfect man. However, the gospel accounts tell us that at the point of his birth, he was already a Saviour, "Christ the Lord" - Luke 2:11.
It is, quite simply, denial of the Word of God being God, in the beginning, so that this Word made everything that was made, that leads some to think it must have been the baptism of the man, Jesus, that marked a significant point of God's approval and divine authority and enabling. Yet at his baptism, the Father declared that he was already well pleased with his Son, before his ministry had even started! Mark 1:11 has God speaking in the past tense, "Thou art my Son, the Beloved, in whom I did delight" (NLT).
So, when you ask, "Why must the hypostatic union be viewed as taking place at Jesus' conception?" it is because of who this One already was, prior to the virgin conceiving, and the Holy Spirit's role in that miraculous conception which brought human nature to meet and unite with the One who made everything that was made (John 1:1-14).