(With the exception of the LDS and the Jehovah's Wtnesses, IIRC, because they do not believe in the Trinity and thus their Baptism cannot, by definition, be intended to be baptism in the name of the Trinity.)
In fact, any person, even a non-baptized person, can administer baptism in the case of an emergency, e.g. a military medic baptizing his mortally wounded fellow soldier, at the dying man's request.
It's in the Catechism. They could have looked it up.
This is a humanistically-derived, errant opinion, not worthy of a clear literal/grammatical/historical/cultural hermeneutic (interpretation) of the command of Christ as found at the end of the Gospel of Matthew, nor as it is applied in the greater context.
The correct interpretation is that this baptism is a rite of initiation (similar to the swearing-in ceremony of induction into our armed forces) into the life-long occupation of disciple-making, by conferring delegated authority under Christ's command.
The water-baptism is not an agency of conferring forgiveness of sins, as many think, the dispensation of which they like to put themselves in command (simony). No, it is a rite performed on the basis of sin and sins already convicted of by the Holy Ghost, confessed and abandoned by the believer, abandoned and forgotten by The Just And Righteous Father--a prerequisite for having been spiritually cleansed by the Blood of The Christ, justified by His resurrection, and reconciled by Him as our Eternal Priest--in preparation for the believer to be accepted into the disciplizing paradigm by water-baptism immersion and concurrent proclamation of The Faith.
This water-baptism is meant solely for the commissioning of regenerated believer-disciples who have already shown their clear devotion to following Christ through being discipled into a saving knowledge of and passion for The Faith, thus committing themselves into being progressively led into spiritual maturity. This baptism (not one of the seven other baptisms mentioned in the Bible) is to be entered by immersion. Figuratively, it is a baptism of total submersion into a death-producing medium, signifying a death to self, to Sin as a Master, and to the world; from which one arises into a life-giving medium as eternally and fully committed to The Christ Master as a bond-slave and disciple-maker.
The meaning and intention of this baptism as an induction/ordination rite was explicitly understood by the disciple/apostles. Any other proposed use would be a misuse and a demeaning of its purpose.
This baptism clearly is to be administered only by an individual who has agreed with the doctrine, submitted himself to this ordination, and who has begun the preparation for disciple-making--one who has demonstrated the intent to bear fruit. That is, though the fruit (not "fruits") of the Spirit are seen in an individual (figuratively) as the observed gradual development of the character of Christ; in contrast, the fruit of the disciple--another and different fruit--is more disciples (learners)(literally) which he has been empowered to recruit and instruct.
Thus the baptism spoken of is not a transaction to be entered into by a new convert, nor by an unregenerated believer, nor by an individual self-proclaiming conversion, and most certainly not by a newborn infant child nor rank unbeliever. The purpose is not salvatory for them.
The agreement described in the article about the nature, purpose, or performance of water-baptism among various extra-Biblical denominations is the gross error of lack of understanding of the mind of Christ by uncommissioned groups falsely proclaiming themselves as appointed purveyors of The Faith, and deceiving seekers of the rudiments of The Faith thereby.
That is a Scriptural view in which I have been instructed, and mine also, of an answer to the incorrect description you have given of carrying out this glorious commandment.
Let Scripture alone be the divider of truth from error.
Look up all the references.