The testimony of Scripture is otherwise:
Jn 5:23, 36 - the Father sent the Son;
God the Son is subject to the Father
That is the nature of the Son of God. He's not only sent by the Father,
He is begotten from the Father. He is the Word and the Image of the Father.
That is not submission, in divinity, that is equality.
In regard to His humanity, becoming the Son of Man, yes, the Son took on submission just as His Father took on authority.
Jn 5:43 - Jesus came in the Father's name, acting as the Father's agent, speaking the Father's words (Jn 12:49-50, 14:24), doing the Father's works (Jn 4:34, 10:25, 17:4) and bearing witness to the One who sent him;
God the Son is subject to God the Father
Same as above. The Son became subject to the Father.
But in eternity past, Son and Father are equal, just as one cannot be 'father' without a son, one cannot be 'son' without a father.
Jn 14:26 - the Father will send the Spirit "in my name," to act in the world as Jesus agent, witness (1Jn 5:7), deputy, doing Jesus' will and acting as Jesus' representative and with his authority;
God the Holy Spirit is subject to the Father
Jn 15:26 - the Son will send the Spirit "from the Father";
God the Holy Spirit is subject to God the Son, as well as to God the Father
Jn 16:27 - as the Father sent the Son into the world, so the Son will send the Spirit into the world.
God the Son is subject to God the Father, and God the Holy Spirit is subject to God the Son.
There has always been an order of function and authority among the equal persons of the Trinity.
There has always been an order, and yet all are without beginning. They just are. I Am. That is not
authority and submission, rather that just is. It is Being. God is one I Am. The nature of the Spirit is
of the Father and of the Son. The Spirit is the Spirit of the Father, the Spirit of the Son.
In regard to humanity, there is submission and direction. The Father flows to the Son, the Son with the Father flows out as the Spirit. The Spirit is now not only the Spirit of God, but also the Spirit of a glorified man, the Spirit of Jesus Christ. Incorporating the man Jesus Christ and His humanity as elements/ingredients of the Spirit. The Father and Son, the Triune God, took on this matter of authority and submission, because man failed at it. After Lucifer rebelled to it. God the Father, and the Son of God, through the Son of Man, establish it.