Why must the hypostatic union be viewed as taking place at Jesus' conception?
Because the man assumed by the Logos is His humanity and no one else. The man belongs to Logos. If the union happens at baptism then by nature that man is not divine but merely endowed with grace of adoption to be one of His children.
One Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all ages. ... Who for us men and for our salvation came down from heaven and was incarnate of the Holy Spirit and the Virgin Mary and became man.
Niceano-Constantinopolitan Creed (381)
The Logos became man for us and for our salvation. He took humanity from the Theotokos this is why devotion to her is crucial for proper Christology. She is the birth giver of God the Word. She gave her humanity to the Logos. Without being born of a virgin the Logos must create His own humanity alien from the rest of humanity because His humanity would be newly created and not identical with the fallen nature like the rest of us.
If any one confess not that Emmanuel is in truth God and that the holy Virgin is therefore Mother of God (Theotokos), for she bare after the flesh the Word of God made flesh, be he anathema.
St. Cyril, Twelve Anathemas, 1.
God the Word won't create a fallen humanity for Him to assume because He is not the author of evil or partake in evil. It's crucial for the Logos to assume our nature to redeem us and that is why He was united with our fallen humanity at conception.
What has not been assumed has not been healed
St. Gregory of Nazianzus, Epistle 101:32.
As regards to whether or not the Holy Spirit guarded Jesus from falling into sin, it's worth to note that even though Jesus is filled with the Spirit in His earthly ministry He is incapable to sin because He is a divine person, not divine-humane person.
If any one say that the one Lord Jesus Christ hath been glorified by the Spirit, using His power as though it were another's, and from Him receiving the power of working against unclean spirits and of accomplishing divine signs upon men; and does not rather say that His own is the Spirit, through whom He has wrought the divine signs, be he anathema.
St. Cyril, Twelve Anathemas, 9.
Because our fallen humanity was assumed at the moment of conception the Logos is one of us. If He assumed the man Jesus of Nazareth at later time and not at Jesus' conception then when He assumed Jesus there won't be real union but mere Samosatan adoptionism. By adoptionism the Logos by nature is not identical with us and our nature won't be healed. Only by assuming the man at conception He assumed humanity to Himself and it belongs to Him not someone else. By redeeming that humanity He redeemed us.