I'm not totally clear on where you're coming from, to be honest. The Son's name is Jesus. It seems that you are concerned that we say "in the name of the Son", but do not specify the name of Jesus. You also seem concerned that it is done not in the name of the Son only, but also in the name of the Father and the Holy Ghost. Am I right?
As far as your comment that there is no Christian "priesthood" spoken of in the Bible, that's another subject. I do believe in the priesthood and could discuss it more.
Yes, you say baptized in the name of the Son but do not say his name, leaving the door open to another name, such as Lucifer Son of the morning. I'm fine with also making the name Father and Holy Spirit known. In the bible they baptized people into the name Jesus, then they also said haven't you heard about the Holy Spirit and layed hands on and the Holy Spirit came to them. And the Father is separate from the Son, so they also told people about the Father.
But as you have it it sort of conforms with antichrist teachings, that Jesus was a prophet, not the Son of God. So you say you're priests were commisioned by Jesus (minimized to maybe a prophet or something) to baptize in the name of the Father, Son and Holy Ghost. Which I believe the some say the Trinity is One God.
The Muslims could do so well, In the name of Muhammed I baptize you in the name of Allah. Because the Catholics also teach that the Trinity is One God ..which is confusing if Jesus is not the Father, The Son and the Holy Ghost.
So how do you justify the Christian priesthood? And don't say Revelations where it says we are all priests and kings with Christ because that means everyone in Christ is a priest, and equal to one another in that sense.