The candidate for baptism must first qualify themselves by "repent[ing] of his sins." That's an interesting pre-condition in the Lord's instruction. Until one has determined to abandon their sins, they are not fit for baptism. They first decide to lay things behind, move forward in following the Lord, determined to serve Him. This decision to make a change must come "through your words." Meaning that before someone can repent, they must first learn the conditions for repentance and following the Lord.
This is much like the instructions given by revelation to this dispensation about how baptism is to be performed. We were told, "All those who humble themselves before God, and desire to be baptized, and come forth with broken hearts and contrite spirits, and witness before the church that they have truly repented of all their sins, and are willing to take upon them the name of Jesus Christ, having a determination to serve him to the end, and truly manifest by their works that they have received of the Spirit of Christ unto the remission of their sins, shall be received by baptism into his church." (D&C 20: 37.)
After having made the decision to be baptized with a broken heart and contrite spirit, confessing before the church that you have repented of your sins (or be willing to change and follow Christ), determined to endure to the end, a person receives from the Spirit of Christ a witness that changes their behavior. Their works show they are penitent. At this point a person is ready for baptism. Until then, they are not ready and the ordinance is not appropriate.
Now the instructions in Section 20 are more complete than the abbreviated statement in Christ's instructions to His Nephite disciples, but it is to the same effect. When this pattern is followed, people are converted and follow Christ. Their baptism matters and will change them. When these instructions are not followed, the ordinance is relatively meaningless and people drift off into inactivity. I believe today the numbers evidence that approximately 10% of those who are baptized are actually converted. The rest are just names and numbers used as membership statistics to be reported and proclaimed each April in a worldwide conference.
The Gospel of Christ is quite exact and it works whenever it is tried. It is tried today in about 10% of the cases of those who are baptized by our missionaries.
Perhaps the ordinance ought to be offered to more of our adults as they come to recognize that they may not have actually been prepared to receive the ordinance when given to them. No matter, there's always the Alma exception. (That's when in the course of baptizing someone else, you go ahead and take the covenant yourself. See Mosiah 18: 13-15.) Clearly Alma was baptizing Helam at the time, and added himself for good measure; he (Alma) feeling the need for the ordinance himself. He went ahead and was baptized again for good measure. This seems to be a precedent that would allow for others to do likewise - perhaps when performing a vicarious baptism for the dead. I leave the Alma exception for your own consideration, and will stop short of advocating such a thing. I just notice things and share what I notice. I'm not trying to convince anyone to do anything.
In Christ's instructions, and in Section 20, the heavy lifting of repentance precedes baptism. Then, after determining to change and follow Christ, leaving behind the foolish errors of the past, the person is fit to be baptized. At that point the baptism symbolizes the new life being undertaken. The presence of the Holy Ghost then ratifies the purging of the repentant, now baptized convert. But that comes next in Christ's teaching.