There is nothing there that prevents secession or even addresses it, only accession of states or changes within a state.
That is a false statement. Secession without the consent of Congress was prohibited by:
The Articles of Confederation
Article I. The Stile of this Confederacy shall be “The United States of America.”
Article II. Each state retains its sovereignty, freedom, and independence, and every Power, Jurisdiction, and right, which is not by this confederation expressly delegated to the United States, in Congress assembled.
Article XIII. Every State shall abide by the determination of the united States in congress assembled, on all questions which by this confederation are submitted to them. And the Articles of this confederation shall be inviolably observed by every State, and the union shall be perpetual; nor shall any alteration at any time hereafter be made in any of them; unless such alteration be agreed to in a congress of the united States, and be afterwards confirmed by the legislatures of every State.
And Whereas it hath pleased the Great Governor of the World to incline the hearts of the legislatures we respectively represent in Congress, to approve of, and to authorize us to ratify the said articles of confederation and perpetual union. Know Ye that we the undersigned delegates, by virtue of the power and authority to us given for that purpose, do by these presents, in the name and in behalf of our respective constituents, fully and entirely ratify and confirm each and every of the said articles of confederation and perpetual union, and all and singular the matters and things therein contained: And we do further solemnly plight and engage the faith of our respective constituents, that they shall abide by the determinations of the united States in congress assembled, on all questions, which by the said confederation are submitted to them. And that the articles thereof shall be inviolably observed by the States we respectively represent, and that the union shall be perpetual.
As the above excerpt states, “Every State shall abide by the determination of the united States in congress assembled, on all questions which by this confederation are submitted to them. And the Articles of this confederation shall be inviolably observed by every State, and the union shall be perpetual; nor shall any alteration at any time hereafter be made in any of them; unless such alteration be agreed to in a congress of the united States, and be afterwards confirmed by the legislatures of every State.” Under the authority of the Articles of Confederation the States engaged “in the name and in behalf of our respective constituents, fully and entirely ratify and confirm each and every of the said articles of confederation and perpetual union” and committed the same to the oath “And we do further solemnly plight and engage the faith of our respective constituents, that they shall abide by the determinations of the united States in congress assembled, on all questions, which by the said confederation are submitted to them. And that the articles thereof shall be inviolably observed by the States we respectively represent, and that the union shall be perpetual.”
The legal definition of “perpetual” is never ceasing, continuous, enduring, lasting, unlimited in respect of time, continuing without intermission or interval (Black's Law Dictionary). A Union is a joinder of separate entities (Black's Law Dictionary). A perpetual Union is therefore “expressly delegated to the United States, in Congress assembled “ as a (federal) joinder of separate entities (States) never ceasing, continuous, enduring, lasting, and unlimited in respect of time; unless such alteration be agreed to in a congress of the united States, and be afterwards confirmed by the legislatures of every State.”
The proponents of unilateral secession then proceed to raise objections to the authority of the Articles of Confederation, but such objections are invalidated by the Constitution and the general principles of the law of nations.
Constitution of the United States
Article. VI.
All Debts contracted and Engagements entered into, before the Adoption of this Constitution, shall be as valid against the United States under this Constitution, as under the Confederation.
This Constitution, and the Laws of the United States which shall be made in Pursuance thereof; and all Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.
The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.
The legal definition of an “Engagement” is “a contract or agreement characterized by an exchange of mutual promises; e.g. engagement to marry (Black's Law Dictionary).
As seen above in the Articles of Confederation, the States entered into an agreement giving their mutual promises: “we do further solemnly plight and engage the faith of our respective constituents, that they shall abide by the determinations of the united States in congress assembled, on all questions, which by the said confederation are submitted to them. And that the articles thereof shall be inviolably observed by the States we respectively represent, and that the union shall be perpetual.” The Constitution then confirmed “All Debts contracted and Engagements entered into, before the Adoption of this Constitution, shall be as valid against the United States under this Constitution, as under the Confederation.” Consequently, the Constitution is very explicit in stating the Union shall be as perpetual as it was with the Engagement of the States under the Articles of Confederation. Any alteration of this perpetual Union therefore requires the consent of the Congress assembled as stated in the Articles of Confederation: And the Articles of this confederation shall be inviolably observed by every State, and the union shall be perpetual; nor shall any alteration at any time hereafter be made in any of them; unless such alteration be agreed to in a congress of the united States, and be afterwards confirmed by the legislatures of every State.”
While the “Constitution Article. VI. All Debts contracted and Engagements entered into, before the Adoption of this Constitution, shall be as valid against the United States under this Constitution, as under the Confederation,” states the Engagement in a perpetual Union “shall be as valid against the United States under this Constitution, as under the Confederation,” it furthermore states, “Constitution of the United States, Article. I.,
Section. 8. The Congress shall have Power [....] To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions; [....] To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.”
One of those powers delegated by the Constitution to Congress is:
“Constitution of the United States. Article. IV.
Section. 3. The Congress shall have Power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States; and nothing in this Constitution shall be so construed as to Prejudice any Claims of the United States, or of any particular State.”
The unilateral secession of a State whose Engagement in the Articles of Confederation and the Constitution faithfully promised “the union shall be perpetual” constitutes the “Prejudice any Claims of the United States” had with respect to “rights and privileges” the perpetual Union of the United States to exercise the “Power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States.”
Such a unilateral secession of a State from its Engagement in the perpetual Union of the United States also constitutes a Prejudice of a number of other powers delegated to the United States and the other States in the Union by the Constitution, not the least of which is:
Constitution of the United States.
Section. 8.
The Congress shall have Power
To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions; [....]
So, the Constitution contains numerous powers delegated to the United States which are Prejudiced by a State acting to unilaterally secede in rebellion and thereby Prejudice the rights, duties, and obligations of the United States and its member States.