Of course, the text says nothing of the sort. For the Nephties, it was a big deal just to have a few missionaries go to the Lamanites. Nothing is ever mentioned about going to "much of the earth."
The rationale in the blog was that in the Old World, Israel was the crossroad of civilizations so the Gospel could spread from that point. "The formulaic phrase 'all nations, kindreds, tongues, and people' occurs 16 times in the text exclusive of the testimonies of the witnesses and Moroni's instructions to the prophet Joseph. The Book of Mormon writers cared about wide dissemination. In pre-Colombian times Mesoamerica was the crossroads of the western hemisphere."
However, it's an inapt comparison. To the extent the Book of Mormon writers "cared about wide dissemination," it was in reference to the latter days when the gospel would be restored and Israel gathered.
I've listed these references below so you can see they were primarily in relation to the last days, not the ancient days, and they are being fulfilled in our day, as prophesied, primarily from the promised land where Nephi lived and wrote these prophesies; i.e., the United States of America. The exceptions, noted below, had nothing to do with the Nephites taking the gospel to all the world.
Think about this carefully. The Book of Mormon writers did care about taking the gospel to all nations, kindreds, tongues and people--but they knew it was not going to happen in their day. Instead, they saw it happening in the future, after the gospel would be restored.
Where was the gospel restored? In the United States.
From where is the gospel being taken to all nations, kindreds, tongues and people? From the United States.
No one can seriously claim that Mesoamerica is sending the gospel to all the world. As wonderful as the missionaries from that area are, they are relatively few in number and mostly supported by the Saints in the United States anyway.
Another assertion in the blog post is the idea that the "blood of Israel" was spread out from Mesoamerica or Central America. We all know that the indigenous people in Mesoamerica, according to the DNA, are Asians. The only recognizable "blood of Israel" in the ancient Americas, according to DNA, was among the indigenous people in North America, primarily around the Great Lakes and New York area. While I agree we cannot prove or disprove the Book of Mormon by DNA, there is no evidence to support the claim that the "blood of Israel" was ever found in Mesoamerica. To claim it spread out from there is, well, you figure it out. To the extent the "blood of Israel" exists at all in ancient Mesoamerica, it would have been brought back after the Mayans returned to Mesoamerica following their migration northward after Mayan civilizations collapsed around 800 A.D. If I understand correctly, the Central American theory claims that Lehi landed in the midst of Mayan civilization (never mentioned in the text), that the Nephites were completely absorbed into the larger culture (also never mentioned in the text) so that their "blood of Israel" was diluted to the point where it is no longer detectable. That's not an irrational explanation; in fact, it's the only explanation that can be made if one believes in the Central American setting.
Back to the text. From the Book of Mormon, we have a string of prophecies about the gospel being taken to the world in the latter day:
1 Nephi 5:18
18 That these plates of brass should go forth unto all nations, kindreds, tongues, and people who were of his seed.
1 Nephi 11:36
36 And it came to pass that I saw and bear record, that the great and spacious building was the pride of the world; and it fell, and the fall thereof was exceedingly great. And the angel of the Lord spake unto me again, saying: Thus shall be the destruction of all nations, kindreds, tongues, and people, that shall fight against the twelve apostles of the Lamb.
1 Nephi 14:11
11 And it came to pass that I looked and beheld the whore of all the earth, and she sat upon many waters; and she had dominion over all the earth, among all nations, kindreds, tongues, and people.
1 Nephi 19:17
17 Yea, and all the earth shall see the salvation of the Lord, saith the prophet; every nation, kindred, tongue and people shall be blessed.
1 Nephi 22:28
28 But, behold, all nations, kindreds, tongues, and people shall dwell safely in the Holy One of Israel if it so be that they will repent.
2 Nephi 26:13
13 And that he manifesteth himself unto all those who believe in him, by the power of the Holy Ghost; yea, unto every nation, kindred, tongue, and people, working mighty miracles, signs, and wonders, among the children of men according to their faith.
2 Nephi 30:8
8 And it shall come to pass that the Lord God shall commence his work among all nations, kindreds, tongues, and people, to bring about the restoration of his people upon the earth.
1 And now, it came to pass that after Abinadi had spoken these words he stretched forth his hand and said: The time shall come when all shall see the salvation of the Lord; when every nation, kindred, tongue, and people shall see eye to eye and shall confess before God that his judgments are just.
25 And the Lord said unto me: Marvel not that all mankind, yea, men and women, all nations, kindreds, tongues and people, must be born again; yea, born of God, changed from their carnal and fallen state, to a state of righteousness, being redeemed of God, becoming his sons and daughters; [this refers to the universal nature of the gospel, not missionary work]
20 Yea, after having been such a highly favored people of the Lord; yea, after having been favored above every other nation, kindred, tongue, or people; after having had all things made known unto them, according to their desires, and their faith, and prayers, of that which has been, and which is, and which is to come; [this refers to how the Nephites were blessed above every other nation and has nothing to do with missionary work]
3 Nephi 26:4
4 And even unto the great and last day, when all people, and all kindreds, and all nations and tongues shall stand before God, to be judged of their works, whether they be good or whether they be evil—
3 Nephi 28:29
29 And it shall come to pass, when the Lord seeth fit in his wisdom that they shall minister unto all the scattered tribes of Israel, and unto all nations, kindreds, tongues and people, and shall bring out of them unto Jesus many souls, that their desire may be fulfilled, and also because of the convincing power of God which is in them.
The other scriptures also use the phrase in this context about the latter days:
Doctrine and Covenants 10:51
51 Yea, that it might be free unto all of whatsoever nation, kindred, tongue, or people they may be.
[This one is critical. The Lord tells Joseph that the Book of Mormon was the fulfillment of the promise to the prophets and disciples who prayed that the gospel would come unto their brethren the Lamanites in the future. "Now, this is not all--their faith in their prayers was that this gospel should be made known also, if it were possible that other nations should possess this land; and thus they did leave a blessing upon this land in their prayers, that whosoever should believe in this gospel in this land might have eternal life; yeah, that it might be free unto all of whatsoever nation, kindred, tongue or people they may be." This revelation was given in Harmony, Pennsylvania, not Quirigua or Tikal. Other nations have possessed the land Nephi referred to, including the Indian nations, the British, the French, the Spanish, and eventually the United States. No land in all the world is a melting pot like the United States. As the revelation describes it, "this land" was to be "free" unto "whatsoever nation, kindred, tongue or people they may be."]
Doctrine and Covenants 42:58
58 And I give unto you a commandment that then ye shall teach them unto all men; for they shall be taught unto all nations, kindreds, tongues and people.
Doctrine and Covenants 112:1
1 Verily thus saith the Lord unto you my servant Thomas: I have heard thy prayers; and thine alms have come up as a memorial before me, in behalf of those, thy brethren, who were chosen to bear testimony of my name and to send it abroad among all nations, kindreds, tongues, and people, and ordained through the instrumentality of my servants.
9 After this I beheld, and, lo, a great multitude, which no man could number, of all nations, and kindreds, and people, and tongues, stood before the throne, and before the Lamb, clothed with white robes, and palms in their hands;
7 And it was given unto him to make war with the saints, and to overcome them: and power was given him over all kindreds, and tongues, and nations.
The testimonies of Joseph Smith and the witnesses verify the realization of the prophecies in the latter days. The gospel was restored in the United States and from there it is being taken to all nations, kindreds, tongues and people.
Joseph Smith—History 1:33
33 He called me by name, and said unto me that he was a messenger sent from the presence of God to me, and that his name was Moroni; that God had a work for me to do; and that my name should be had for good and evil among all nations, kindreds, and tongues, or that it should be both good and evil spoken of among all people.
The Testimony of Eight Witnesses
Be it known unto all nations, kindreds, tongues, and people, unto whom this work shall come: That Joseph Smith, Jun., the translator of this work, has shown unto us the plates of which hath been spoken, which have the appearance of gold; and as many ...
The Testimony of Three Witnesses
Be it known unto all nations, kindreds, tongues, and people, unto whom this work shall come: That we, through the grace of God the Father, and our Lord Jesus Christ, have seen the plates which contain this record, which is a record of the people of ...
When we read the text, we can read it through any of several filters, but ideally we would read it without a particular lens over our eyes and minds.