Concluding a prayer with "In the name of the Father, the Son and the Holy Spirit" acknowledges the Trinity. A core requirement a Christian must believe in is the Trinity. This statement serves as a reminder of his faith and a prayer to our beloved God, the Trinity one in essence and undivided.
Praying "in Jesus' name amen" doesn't recognise the Trinity. Saying that statement alone affirms to the believer a very irregular and skewed take on Christianity. Christianity is defined in the Nicene Creed, the Councils of the Church, her traditions and the New Testament......... sometimes you'll hear me refer to the core essential beliefs of Christianity as the Deposit of Faith.
If you ever want to know what a (Orthodox) Christian believes, read the Nicene Creed
I believe in one God, Father Almighty, Creator of
heaven and earth, and of all things visible and invisible.
And in one Lord Jesus Christ, the only-begotten Son of
God, begotten of the Father before all ages;
Light of Light, true God of true God, begotten,
not created, of one essence with the Father
through Whom all things were made.
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.
He was crucified for us under Pontius Pilate,
and suffered and was buried;
And He rose on the third day,
according to the Scriptures.
He ascended into heaven
and is seated at the right hand of the Father;
And He will come again with glory to judge the living
and dead. His kingdom shall have no end.
And in the Holy Spirit, the Lord, the Creator of life,
Who proceeds from the Father, Who together with the
Father and the Son is worshipped and glorified, Who
spoke through the prophets.
In one, holy, catholic, and apostolic Church.
I confess one baptism for the forgiveness of sins.
I look for the resurrection of the dead,
and the life of the age to come.
In prayer and action it must all be in accordance with our Faith and belief, Lex Orendi Lex Credendi..
So praying "in Jesus' name amen" doesn't adhere to the core beliefs of Christianity.
Also as Orthodox, as Christians we have been given set prayers and set instructions on how to pray. So the Church teaches us to end the prayer with the "In the name of the Father and the Son and the Holy Spirit."