:
You are right, the story of the baptism of Christ does squash the idea that the Trinity is modalistic.
But the story of the baptism also brings up concerns for the Trinity. Let me explain:
You say that in the story of the baptism, "the 3 PERSONS of the Triity all apear independently doing something in the same place at the same time".
You are right about at the same "time", but you are not right about in the same "place".
For instance: Jesus was on the earth coming up out of the waters of the river Jordan(1), God the Father was in the heavens, telling everyone that "this is my beloved Son, in whom I am well please"(2), and the HS was in between the Father in the heavens and the Son on the earth, but eventually did fall upon Jesus on the earth, but there was a time when he had not fallen
upon him (3).
So you see, Jesus and God the Father and the HS, were indeed there at the same time, but they surely were not in the same place at the same time. The 3 were indeed independent, and were at different locations at the same time.
God the Father in heaven.
Jesus on earth.
HS in between God the Father and Jesus.
So the story of the baptism is tough on the Trinity concept, and vindicates JS.
That is a bit of a reach to get out of the plan picture of one person speaking, one being baptized and one appearing as a dove.
Dicache,
Chapter 7. Concerning Baptism
And concerning
baptism,
baptize this way: Having first said all these things,
baptize into the name of the
Father, and of the
Son, and of the
Holy Spirit,
Matthew 28:19 in living water. But if you have not living water,
baptize into other water; and if you can not in cold, in warm. But if you have not either, pour out water thrice upon the head into the name of Father and Son and
Holy Spirit. But before the
baptism let the baptizer fast, and the
baptized, and whatever others can; but you shall order the
baptized to fast one or two days before.
The Fathers:
St. Clement of Rome, Letter to the Corinthians (C. 98 A.D.):
"Do we not have one God, one Christ, and one Spirit of Grace poured out upon us? And is there not one calling in Christ?"
The Martyrdom of St. Polycarp (C. 155 - 157 A.D.):
"...In this way and for all things I do praise you, I do bless you, I do glorify you through the eternal and heavenly High Priest Jesus Christ, your beloved child: through whom be glory to you with Him and with the Holy Spirit, both now and through ages yet to come. Amen."
St. Theophilus of Antioch, To Autolycus (181 A.D.):
"The three days before the luminaries were created are types of the Trinity: God, His Word, and His Wisdom."
St. Irenaeus of Lyons, Against Heresies (C. 180 A.D.):
"If any one, therefore says to us, ‘How then was the Son produced by the Father?’ we reply to him, that no man understands that production, or generation, or calling, or revelation, or by whatever name one may describe His generation, which is in fact altogether indescribable. Neither Valentinus, nor Marcion, nor Saturninus, nor Basilides, nor angels, not archangels, nor principalities, nor powers (possess this knowledge), but the Father only who begat, and the Son who was begotten. Since therefore His generation is unspeakable, those who strive to set forth generations and productions cannot be in their right mind, inasmuch as they undertake to describe things that are indescribable."
St. Ambrose of Milan, Hexameron (Post 389 A.D.):
"But let us consider the course of our own creation. He says: ‘Let Us make man to our image and to our likeness.’ Who says this? Is it not God, who made you?...To whom does He say it? Certainly not to Himself, for He does not say ‘Let Me make’ but ‘Let Us make.’ Nor to the Angels, for they are ministers; and servants can have no partnership in the operation of the master, nor works with their author. It is the Son to whom He speaks, even if the Jews will not have it and the Arians fight against it...[And it is the Son] who is the image of God the Father, the Son who always is and who was in the beginning."
Catholic Apologetics - Lumen Verum part 1