The Militia turns out to be an difficult subject. The first thing I need to point out is that what is said about the legislature and the militia is part of a section starting out with the words, "The Congress shall have the Power to". Note: this is a grant of power, NOT an assignment of responsibility. Secondly, this section includes BOTH, funding the Army AND funding the Militia. While these two powers are listed separately, and the duties of the militia are specifically listed, nowhere does it say that the two needs can't be combined. Finally, the only thing clearly given to the states is appointment of Officers and the authority to train.
With these thoughts in mind, lets look at the words of the actual Constitution. It does clearly say that Congress, not the President, not the Govenor, has the authority to call out the Militia. And it says when that is done, it shall be to execute the Laws of the Union, suppress Insurrections and repel Invasions.
It also does clearly refer to the militia as "of the several states". That may be a potential future arguing point, but as you will see shortly a lot of water has already gone over the dam. It is time to look at the history.
The first thing you should note from the screen is that Congress has been fiddling with the subject since 1902. The most important thing you should notice is that we have created a federal guard to mirror the state guard, and we have declared that all recruits to the state guard are automatically also recruits to the federal guard. Finally we have decided that the federal guard can call up the state guard at any time in support of the needs of the regular army.
It seems to me, from a Constitutional perspective, that the idea of a Militia with the responsibility to "execute the Laws of the Union, suppress Insurrections and repel Invasions" has gotten completely lost within the emphasis on merging the State guard with the National Guard. This question WAS brought to the Supreme Court as far back as 1918 with Cox v Wood, and the decision was that the power to raise armies was "not qualified or restricted by the provisions of the militia clause."
The idea that the needs of the regular army supercedes any Militia needs was further emphasized in 1990 with Perpich v. DOD. Those two cases are the total Supreme Court involvement in the Militia. No case has been brought that I can see questioning the need/constitutionality of nationalizing the guard for internal purposes of "executing the Laws of the Union, suppressing Insurrections and repelling Invasions". And without the nationalization of the guard it seems hard to justify giving the President any role in the process.
The President is currently doing his best to silence any questions on that score.

The Militia

Constitution

Article I, Section 8
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;

Article II, Section 2
The President shall be Commander in Chief of the Army and Navy of the United States, and of the Militia of the several States, when called into the actual Service of the United States;

Legislative Actions

1902 Dick Act - Provided federal funding and training for National Guard units. Contemplated that national guard forces would be used only in the United States

1908 - Amended to allow National Guard use within or without of the country.

1916 - Every Guardsman became a member of BOTH the State Guard and the National Guard. And the President had the authority to call the state guard into federal service.

1933 - Clarified and emphasized the 1916 law. Calling the Guard into federal service was to be done by Congress and required a national emergency needing troops in excess of Regular Army

1952 - Expanded callup to include both active duty and active duty for training. Governor consents were sought and routinely obtained

1985 - removed governors right to object to their guard being called up

Supreme Court interpretations

Cox v. Wood, 247 U. S. 3 (1918), the plenary power to raise armies was "not qualified or restricted by the provisions of the militia clause."

Perpich v. DOD, 496 U.S. 334 (1990), Article I's plain language, read as a whole, establishes that Congress may authorize members of the National Guard of the United States to be ordered to active federal duty for purposes of training outside the United States without either the consent of a state governor or the declaration of a national emergency. Pp. 496 U. S. 347-355.