Why don't they come out and make a public declaration? They were specifically prohibited from doing so.
In the same chapter cited in the OP (3 Nephi 28), it is said that the 3 Nephites were commanded not to share certain information:
16 But it came to pass that they did again minister upon the face of the earth; nevertheless they did not minister of the things which they had heard and seen, because of the commandment which was given them in heaven. (see also vss. 13-14)
Their identity was not to be widely made known to the world: "for they are hid from the world." (see vs. 25)
The chapter under consideration (with respect to the 3 Nephites) indicates that they are not at liberty to do as described in the OP. If John's commission is the same/similar, it is reasonable to believe that he was given similar instructions.
If some guy came out and claimed to be John the Apostle, why would the world take him more seriously than One who came out and claimed to be the Son of God 2,000 years ago? Or a prophet who brought plagues and parted the sea ~3400 years ago? Or any of God's messengers over the years who were mocked and rejected? Jesus very bluntly identified this feature of human nature in Luke 16:
31 And he said unto him, If they hear not Moses and the prophets, neither will they be persuaded, though one rose from the dead.
Presumably the same could be said for one who returned after having been thought to be dead.
Various people have in fact come out and claimed to be such a person, or to have met them, and they were taken about as seriously as you would predict. The ministry of John & the 3 Nephites was designed to quietly go about doing good, not score headlines.
My own experience is that conversion is a very personal, individual process, not a group experience. I suspect John & the 3 Nephites spend a lot more time, like the good shepherd, going after the sheep that went astray, than they do trying to score applause from the 99 (or anyone else). I do hope their memoirs are released some day--the stories they could tell would surely be fascinating.
This is but a subset of the larger question "why doesn't God just show Himself?"
We could run endless variations on the same theme, where God could presumably put evidence in front of people that would be so overwhelming nobody could possibly disbelieve. Cleary He could do that. But He doesn't. Why? This is a question not just for Latter-day Saints, but for all Christians, and most Theists.
I offer a response to this conundrum in this video on my channel: Aviation & Salvation: The Parable of the Solo Flight.
In short, if God personally appeared to every person, showed them what eternal life looks like, showed them what damnation looks like, and commanded them with thunder and lightning to obey His commandments—rates of obedience would probably go up, but they would go up for the wrong reason. There would be belief without faith, change without commitment, participation without worship, and obedience without transformation.
I believe God wants more than just people’s actions, He wants their hearts. He doesn’t just want people to walk the strait & narrow; He wants to change people in the process.
God wants us to learn to trust, and specifically to learn to trust in His Beloved Son, Jesus Christ. If He gave us literally all the information He has it would both blow our minds and destroy the exercise of learning to trust. He mercifully does not give us information we are not ready for--if He did, it would only serve to condemn us further (see Luke 12:47-48, Doctrine & Covenants 82:3)
I suggest that John & the 3 Nephites have not publicly identified themselves (let alone used the headlines that would create) to evangelize because:
- It wouldn't convince people who do not want to believe. John 6:66 pretty well shows that miracles do not produce enduring belief.
- It is at odds with the way God does His work--He does not compel people to disbelieve on pain of irrationality. It would also be at odds with the very purpose of this phase of the Plan of Salvation, described in greater detail in the video linked above.
- Obtaining more knowledge than they are ready for doesn't help people, it makes them more accountable, and more guilty if they disobey. God is a good parent who gives people truth line upon line, precept upon precept, here a little and there a little.
Appendix--the death of John
There have been many Christians over the years who have interpreted John 21:22-23 as a statement that John the Son of Zebedee would never die (this certainly isn't the only way these verses have been interpreted, but it's a common interpretation). This belief has been more and more rejected in recent generations under the influence of naturalistic thought, but for centuries many Christians believed this and did not consider it irrational.
For any who are interested, we do not in fact have any early evidence of John's death. There are a variety of competing traditions about his demise, or lack thereof. The early church couldn't even agree on where his body had been buried. Let's consider the writers who were in the best position to know about John's death:
- Polycarp of Smyrna (knew John personally) - says nothing on the topic
- Ignatius of Antioch (knew John personally, according to Jerome's Chronicon) - says nothing on the topic
- Papias of Hieropolis (knew John personally) - most of his writings have been lost; from what we have, he says nothing on the topic
- Irenaeus of Lyons (disciple of Polycarp, who knew John personally) - Irenaeus grew up in the shadow of John's ministry, in the region he had led, taught by people he had taught, and he is the earliest writer (whose work survives) to discuss the fate of John, and while Irenaues says John remained in Ephesus until the time of Trajan, Irenaues in fact says nothing at all about John's death.
This evidence is not dispositive, but it is curious. I am quite comfortable with the possibility that John, like the 3 Nephites, reached a point where his public ministry came to a close (circa AD 100), that he underwent a transfiguration such that his body would not decay or die, and that he has done God's work without fanfare for many years since.