Not according to the FAQ at the Texas Guard Web site.
https://tmd.texas.gov/texas-state-guard-faq
US Constitution, Art. 1, Section 8. Powers of Congress
To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions;
To provide for organizing, arming, and disciplining, the Militia, and for governing such Part of them as may be employed in the Service of the United States, reserving to the States respectively, the Appointment of the Officers, and the Authority of training the Militia according to the discipline prescribed by Congress;
From the TXSG website you cited: The Texas State Guard (TXSG) is governed by Title 32 of the United States Code as a State Defense Force only. As such, the Governor is our Commander in Chief and we work solely at his direction. The Army National Guard may be activated to serve with the U.S. Army and the Air National Guard may be activated to serve with the U.S. Air Force, but the Texas State Guard would never be activated to serve with federal military forces.(Emphasis added)
SO, you are correct as to what the TXSG website says. Let's look at Title 32.
32 U.S. Code - National Guard
§ 109. Maintenance of other troops
(a) In time of peace, a State, the Commonwealth of Puerto Rico, the District of Columbia, Guam, or the Virgin Islands may maintain no troops other than those of its National Guard and defense forces authorized by subsection (c).
...
(c) In addition to its National Guard, if any, a State, the Commonwealth of Puerto Rico, the District of Columbia, Guam, or the Virgin Islands may, as provided by its laws, organize and maintain defense forces. A defense force established under this section may be used within the jurisdiction concerned, as its chief executive (or commanding general in the case of the District of Columbia) considers necessary, but it may not be called, ordered, or drafted into the armed forces.
Thus Congress has used its power to authorize the creation of State Defense Forces, that may not be called, ordered, or drafted into the armed forces.
What are the "armed forces" into which the State Defense Forces may not be called, ordered, or drafted?
32 U.S. Code - National Guard
§ 101. Definitions
(2) “Armed forces” means the Army, Navy, Air Force, Marine Corps, and Coast Guard.
(3) “National Guard” means the Army National Guard and the Air National Guard.
(4) “Army National Guard” means that part of the organized militia of the several States and Territories, Puerto Rico, and the District of Columbia, active and inactive, that--
(A) is a land force;
(B) is trained, and has its officers appointed, under the sixteenth clause of section 8, article I, of the Constitution;
(C) is organized, armed, and equipped wholly or partly at Federal expense; and
(D) is federally recognized.
The Texas State Guard is a land force and is trained, and has its officers appointed, under the sixteenth clause of section 8, article I, of the Constitution, but is neither organized, armed, and equipped wholly or partly at Federal expense, nor federally recognized.
Per 32 § 109 it cannot be called, ordered, or drafted into the armed forces by President. But Congress has merely kept the President from mobilizing State Defense Forces into the Army, Navy, Air Force, Marine Corps, and Coast Guard. It has not forbidden the President from calling out State Defense Forces into Federal service to repel invasion, suppress insurrection, or enforce the laws of the United States.
In that case, they would be State Militia on Federal Service. That has not happened since the early days of WW2, when FDR called the Territorial Militias of Alaska and Hawaii into Federal Service, and they served as active duty militia, in Federal Service, and not as members of the Army, Navy, Air Force,Marine Corps, and Coast Guard.
Would it be a mess, given that, at present, Congress has repealed the portions of the old Militia Act that provided a means to pay State Militia called into Federal Service? Absolutely.
Is this a purely theoretical issue, practically speaking? Again yes. What practical use would State Defense Forces, whom the President has no present legal authority to pay, and may use only to repel invasion, suppress insurrection, or enforce the laws of the United States be if the President called them into Federal Service?
But theoretical issues sometimes become real. The Constitution of the Soviet Union, in a dig at the United States, guaranteed the right of each of its 15 member Republics to leave anytime. That was far more theoretical, as everyone knew the Russian Communist Party would never allow a Republic to leave.
Then 1991 happened, and they all left. I hope they does not succeed, but Biden, and his handlers, are working very hard to bring about the sort of crisis where purely theoretical issue become very real indeed.
Thanks.
That begs the question. If Biden tried to Federalize them, and they refused to obey, or Abbott objects, then what?
It would be interesting if it was not so serious.