What does the Tenth Amendment have to do with this?

Everything. The Tenth Amendment is the rule that turns the Constitution's silence on a state leaving into a reserved right of the states and the people.

What it says

The Tenth Amendment reads in full: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people." It is one sentence. It is the closing argument of the Bill of Rights, and it sets the default rule for the entire system. Anything not handed to Washington stays with the states, or with the people themselves.

Connect it to the silence and the answer writes itself

Walk the logic one step at a time. The Constitution is a list of delegated powers. The power to prevent a state from leaving is not on that list. The Tenth Amendment says every power not on the list is reserved to the states or the people. Therefore the decision about whether to remain in the union is a reserved power. It belongs to Texas and to Texans, not to the federal government. There is no clever step hidden in that chain. It is just the Tenth Amendment applied to a power Washington was never given.

The Supreme Court itself has described the amendment exactly this way

In United States v. Darby (1941), the Court explained that the Tenth Amendment "states but a truism that all is retained which has not been surrendered." That is not a movement slogan. That is the Supreme Court describing the amendment as a rule of retained sovereignty: the states kept everything they did not give up. The power to hold a state in the union by force was never surrendered, because it was never granted, so it was retained.

Madison drew the same map

James Madison, in Federalist No. 45, told the ratifying public precisely how the balance would work. As he wrote, "The powers delegated by the proposed Constitution to the federal government are few and defined. Those which are to remain in the State governments are numerous and indefinite." Few and defined on one side, numerous and indefinite on the other. The federal government was meant to be the one with the short, specific list. Jefferson made the stakes plain: to step beyond the boundaries drawn around the powers of Congress, he warned, "is to take possession of a boundless field of power, no longer susceptible of any definition."

Forcing a state to stay is the ultimate version of what the Court has already forbidden

The modern Court has built a whole doctrine, called anti-commandeering, holding that the federal government cannot conscript a state's government to carry out federal aims. In New York v. United States (1992) the Court struck down a law that tried to force states to act as federal instruments. Now apply that principle to its limit. If Washington cannot commandeer a state's legislature to pass a single statute, the claim that it can commandeer a state's entire existence, forcing the whole apparatus of Texas to remain a federal subunit against the will of its people, is the most extreme commandeering imaginable. The smaller version is already unconstitutional.

The bottom line

The Tenth Amendment is the legal hinge. The Constitution never gave Washington the power to keep a state in the union, and the Tenth Amendment says that unsurrendered power stays with the states and the people. The states created the federal government. They retain the authority to decide their own membership. That is not insurrection. It is sovereignty operating exactly as the Constitution describes.