Real quick before I respond. Wouldn't it be understood as "in the name of God." (Fullest revealed name being Jesus) end quote. Or basic english remove prepositional phrases (of the father, of the son and of the holy ghost) stop at name and you still have a complete sentence. Name singular referring to Jesus. Beside the faith doesn't lie in the Titles He holds it lies in His name Jesus.
One can use name and authority of, but there is also an indication that when I say, "In the name of Yeshua," I am speaking under the power of his reputation. It comes from an understanding of YHWH in the Tanakh making a name/reputation for himself in the very splashy work in getting the Israelites out of Egypt. Before that time, YHWH was known only to the Patriarchs (Adam to Noach then through Shem to Abraham, Isaac, and Yacob.)