(All that follows is my opinion.)
Co-equal according to the divine essence is not at all incompatible (and frankly has nothing to do with) with the Son assuming a human essence (which is not equal to the divine essence) in addition to His divine essence and mediating between the Father and humanity with it. If you had actually followed the Athanasian Creed instead of misinterpreting what it was saying you'd know this, for it is in the second portion of it which deals with the Incarnation. See the rest of it:
"He, therefore, who wishes to be saved, must believe thus about the Trinity. It is also necessary for eternal salvation that he believes steadfastly in the incarnation of our Lord Jesus Christ. Thus the right faith is that we believe and confess that our Lord Jesus Christ, the Son of God, is both God and man. As God, He was begotten of the substance of the Father before time; as man, He was born in time of the substance of His Mother. He is perfect God; and He is perfect man, with a rational soul and human flesh. He is equal to the Father in His divinity, but inferior to the Father in His humanity. Although He is God and man, He is not two, but one Christ. And He is one, not because His divinity was changed into flesh, but because His humanity was assumed unto God. He is one, not by a mingling of substances, but by unity of person. As a rational soul and flesh are one man: so God and man are one Christ. [[He died for our salvation, descended into Hell, and rose from the dead on the third day. He ascended into Heaven, sits at the right hand of God the Father almighty.]] From there He shall come to judge the living and the dead. At His coming, all men are to arise with their own bodies; and they are to give an account of their own deeds. Those who have done good deeds will go into eternal life; those who have done evil will go into the everlasting fire."
Notice the part bolding in red, this refers to the events of mediation. How is this known? Let's look at two relevant passages describing it:
Hebrews: "But [[Christ has arrived as a high priest of the good things to come.]] Through the greater and more perfect tent not made by hands, that is, not of this creation, and not by the blood of goats and calves, but [[by his own blood, he entered once for all into the most holy place, obtaining eternal redemption.]] For if the blood of goats and bulls and the ashes of a young cow sprinkled on those who are defiled sanctify them for the ritual purity of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, cleanse our consciences from dead works to serve the living God? And [[because of this, he is the mediator of a new covenant,]] in order that, because a death has taken place for the redemption of transgressions committed during the first covenant, those who are the called may receive the promise of the eternal inheritance."
1 Timothy: "For there is one God and one mediator between God and human beings, the man Christ Jesus, [[who gave himself a ransom for all,]] the testimony at the proper time..."
Now, let's talk about the Athanasian Creed: show me two statements in it that contradict one another, and explain why it is a contradiction. All that is bolded in red is referring to the same events. I have ignored your term "subordination" because you didn't define it.
The fact that you did not use this is why it leads me to believe you did not read it. So again, strictly from the Athanasian Creed: show me two statements that contradict.
(All that is above is my opinion.)