He probably did not mean “Father and Son and Holy Spirit” when he spoke “we”. I have not read that he did. He being the Rev. Andres Arango, who, before his resignation on February 1, 2022, had served parishes in the metropolitan area of Phoenix, Arizona.
What had he meant when he said at parish baptisms: “We baptize you in the name of the Father and of the Son and of the Holy Spirit”? Was he indicating that the word “we” meant a baptism was performed by the members of the Church catholic (that is a lower case “c”) or the local diocese or parish … and he was the appointed speaker and minister at the font?
Was no one following the text for Holy Baptism? Had no one heard this variation in the previous sixteen years?
That last question lands closer to my musing. If I were to alter or revise or change the language of a rite in Trinity Evangelical Lutheran Church, would a member notice? Yes, and for at least several reasons.
The Order of Holy Baptism is printed in the Lutheran Service Book, which is the accepted and standard text for this congregation's practice. The congregation called the pastor into a public office, and thus the office, with its accompanying texts, shape the minister and his preaching and teaching; the text does not concede ground to the pastor. Thus the congregation need not (should have no concern) that the service will be massaged and managed by the pastor’s Saturday or blog-influenced whims. The rites are Church rites, not the domain of the local pastor. Not the man but the office as given and described by the Lord of the Church carries the weight of the words. So let the words of a man decrease and let the Word of Christ increase.
Also, the congregation has been hearing the same rite for decades. And having been catechized year after year with the same language, variations are easily identified. (Do you recall what happened when the text of TLH page 15 was gently revised for a proposed LSB Divine Service version?) A fond example is when one member often mouths “Holy Ghost” in the invocation. (This same person also regularly notes “My mother named me Christian because she wanted me to remember who I am in my baptism”.) The congregation’s confession has been learned and marked and inwardly digested so that if words were to vary, members would notice when that “same taste” is not there. Rites teach; so, teach right.
For the purpose of rightly spoken rites is not pedantic accuracy, but for the comfort of the sinner. Words matter. God’s words matter. God’s words matter because His words make alive, raise the dead, forgive sin, and make coheirs with Christ. And if one alters the words, what is being said or declared? And by whose authority is the gift given? Can “we” bear the weight / glory of the promise? Will the one baptized have the certainty to whom she belongs? So, invalid and valid are not my first step in this discussion. Rather, does the rite confess “in the name of the Father and of the + Son and Holy Spirit”? For those words … those words of the Scriptures breathed out by God … draw us to Himself. He has revealed Himself in the person of His Son, the Word that became flesh and dwelt among us and in Whom the fullness of Deity dwells, the One crucified and risen and ascended; and baptized into the Son, the Holy Spirit moves the new heart and mind and mouth to call out “O Father!” The Scriptures will not tolerate a pagan innovation of —and faith cannot be built on something as amorphous as— “Mother and Child and Sanctifier”.
Our Father desires us to be certain of His promise, our inheritance in Christ Jesus, with the Holy Spirit the Guarantee.
(Black ink is used for the text of what is said; red ink is used for what is done. Hence, "Say the black and do the red.")
Here is an fine article by Gene Veith, "Invalid Baptisms?"