The scenario in the question says that:
- They [3/4ths of citizens] agree that normal processes like public testimony, initiatives, referenda, elections, and recall effort have failed to make any changes that the public wants
- Nobody is suggesting violence or rebellion.
- All processes will follow pre-abolishment laws.
But if "normal processes" cannot be used to make the changes desired, then "pre-abolishment laws" are not being followed in all respects. Moreover, id the current elected officials do not agree to be replaced, then violence of some degree will be needed to remove them, or they may initiate violence in the form of attempted arrests of those leading the change process for violations of those "pre-abolishment laws". So these conditions are inconsistent, they cannot all be true.
Luther v. Borden
There is at least one highly relevant episode in US history, adn it was explored in the Supreme Court case of Luther v. Borden, 48 U.S. 1 (1849) In connection with he opinion the Justia Syllabus says:
At the period of the American Revolution, Rhode Island did not, like the other States, adopt a new constitution, but continued the form of government established by the Charter of Charles the Second, making only such alterations, by acts of the Legislature, as were necessary to adapt it to their condition and rights as an independent State.
But no mode of proceeding was pointed out by which amendments might be made.
In 1841, a portion of the people held meetings and formed associations which resulted in the election of a convention to form a new constitution to be submitted to the people for their adoption or rejection.
This convention framed a constitution, directed a vote to be taken upon it, declared afterwards that it had been adopted and ratified by a majority of the people of the State, and was the paramount law and constitution of Rhode Island.
Under it, elections were held for Governor, members of the Legislature, and other officers, who assembled together in May, 1842, and proceeded to organize the new government.
But the charter government did not acquiesce in these proceedings. On the contrary, it passed stringent laws, and finally passed an act declaring the State under martial law.
In May, 1843, a new constitution, which had been framed by a convention called together by the charter government, went into operation, and has continued ever since.
The question which of the two opposing governments was the legitimate one, viz., the charter government or the government established by the voluntary convention, has not heretofore been regarded as a judicial one in any of the State courts. The political department has always determined whether a proposed constitution or amendment was ratified or not by the people of the State, and the judicial power has followed its decision.
The courts of Rhode Island have decided in favor of the validity of the charter government, and the courts of the United States adopt and follow the decisions of the State courts in questions which concern merely the constitution and laws of the state.
The question whether or not a majority of those persons entitled to suffrage voted to adopt a constitution cannot be settled in a judicial proceeding.
The Constitution of the United States has treated the subject as political in its nature, and placed the power of recognizing a State government in the hands of Congress. Under the existing legislation of Congress, the exercise of this power by courts would be entirely inconsistent with that legislation.
The President of the United States is vested with certain power by an act of Congress, and in this case, he exercised that power by recognizing the charter government.
Although no State could establish a permanent military government, yet it may use its military power to put down an armed insurrection too strong to be controlled by the civil authority. The State must determine for itself what degree of force the crisis demands.
In the Court's formal opinion, Chief Justice Taney wrote:
We do not understand from the argument that the constitution under which the plaintiff acted is supposed to have been in force after the constitution of May, 1843, went into operation. T he contest is confined to the year preceding. The plaintiff contends that the charter government was displaced, and ceased to have any lawful power, after the organization, in May, 1842, of the government which he supported, and although that government never was able to exercise any authority in the State nor to command obedience to its laws or to its officers, yet he insists that it was the lawful and established government upon the ground that it was ratified by a large majority of the male people of the State of the age of twenty-one and upwards, and also by a majority of those who were entitled to vote for general officers under the then existing laws of the State. The fact that it was so ratified was not admitted, and, at the trial in the Circuit Court, he offered to prove it by the production of the original ballots and the original registers of the persons voting, verified by the oaths of the several moderators and clerks of the meetings, and by the testimony of all the persons so voting, and by the said constitution, and also offered in evidence for the same purpose that part of the census of the United States for the year 1840 which applies to Rhode Island and a certificate of the secretary of state of the charter government showing the number of votes polled by the freemen of the State for the ten years then last past.
The Circuit Court rejected this evidence, and instructed the jury that the charter government and laws under which the defendants acted were, at the time the trespass is alleged to have been committed, in full force and effect as the form of government and paramount law of the State, and constituted a justification of the acts of the defendants as set forth in their pleas.
It is this opinion of the Circuit Court that we are now called upon to review. ...
Certainly the question which the plaintiff proposed to raise by the testimony he offered has not heretofore been recognized as a judicial one in any of the State courts. In forming the constitutions of the different States after the Declaration of Independence, and in the various changes and alterations which have since been made, the political department has always determined whether the proposed constitution or amendment was ratified or not by the people of the State, and the judicial power has followed its decision.
The point, then, raised here has been already decided by the courts of Rhode Island. The question relates altogether to the constitution and laws of that State, and the well settled rule in this court is that the courts of the United States adopt and follow the decisions of the State courts in questions which concern merely the constitution and laws of the State.
Besides, if the Circuit Court had entered upon this inquiry, by what rule could it have determined the qualification of voters upon the adoption or rejection of the proposed constitution unless there was some previous law of the State to guide it? It is the province of a court to expound the law, not to make it. And certainly it is no part of the judicial functions of any court of the United States to prescribe the qualification of voters in a State, giving the right to those to whom it is denied by the written and established constitution and laws of the State, or taking it away from those to whom it is given; nor has it the right to determine what political privileges the citizens of a State are entitled to, unless there is an established constitution or law to govern its decision.
he fourth section of the fourth article of the Constitution of the United States provides that the United States shall guarantee to every State in the Union a republican form of government, and shall protect each of them against invasion, and on the application of the legislature or of the executive (when the legislature cannot be convened) against domestic violence.
Under this article of the Constitution, it rests with Congress to decide what government is the established one in a State. For as the United States guarantee to each State a republican government, Congress must necessarily decide what government is established in the State before it can determine whether it is republican or not. And when the senators and representatives of a State are admitted into the councils of the Union, the authority of the government under which they are appointed, as well as its republican character, is recognized by the proper constitutional authority. ...
by the act of February 28, 1795, provided that,
in case of an insurrection in any State against the government thereof, it shall be lawful for the President of the United States, on application of the legislature of such State or of the executive (when the legislature cannot be convened), to call forth such number of the militia of any other State or States, as may be applied for, as he may judge sufficient to sufficient to suppress such insurrection.
By this act, the power of deciding whether the exigency had arisen upon which the government of the United States is bound to interfere is given to the President. He is to act upon the application of the legislature or of the executive, and consequently he must determine what body of men constitute the legislature, and who is the governor, before he can act. The fact that both parties claim the right to the government cannot alter the case, for both cannot be entitled to it. ...
It is true that, in this case, the militia were not called out by the President. But, upon the application of the governor under the charter government, the President recognized him as the executive power of the State, and took measures to call out the militia to support his authority if it should be found necessary for the general government to interfere, and it is admitted in the argument that it was the knowledge of this decision that put an end to the armed opposition to the charter government and prevented any further efforts to establish by force the proposed constitution. The interference of the President, therefore, by announcing his determination was as effectual as if the militia had been assembled under his orders. And it should be equally authoritative. ...
The court thus upheld the verdict for the defendants, and the authority o the government recognized by the President.
It should be noted that this Case arose before the passage of the 14th and 15th amendments to the Federal Constitution, and particularly the Equal protection and Sue process clauses of the 14th, and the "one man, one vote" cases later decided under those clauses. If a state government in 2020 had laws similar to the Charter Government of Rhode Island in 1840, they would be struck down as against those clauses and amendments.
But unless an "abolishment" as described in the question were to be recognized and the resulting government approved by the President and Congress, it would not be lawful, and federal authority and military force could be used to support the lawful government.