I believe this ruling from the Supreme Court OF THE U.S. in 1918 will shed some light on the claims as she mentions this often:Arver v. United States, 245 U.S. 366 (1918
here:
Compelled military service is neither repugnant to a free government nor in conflict with the constitutional guaranties of individual liberty. Indeed, it may not be doubted that the very conception of a just government and its duty to the citizen includes the duty of the citizen to render military service in case of need, and the right of the government to compel it.
here:
Compulsory military service is not contrary to the spirit of democratic institutions, for the Constitution implies equitable distribution of the burdens no less than the privileges of citizenship.
and here:
It may not be doubted that the very conception of a just government and its duty to the citizen includes the reciprocal obligation of the citizen to render military service in case of need and the right to compel it.
and here:
provision of the Pennsylvania constitution of 1776. "That every member of society hath a right to be protected in the enjoyment of life, liberty and property, and therefore is bound to contribute his proportion towards the expense of that protection, and yield his personal service when necessary, or an equivalent thereto." Art. 8, (Thorpe, American Charters, Constitutions and Organic Laws, vol. 5, pp. 3081, 3083.)
and finally here:
Finally, as we are unable to conceive upon what theory the exaction by government from the citizen of the performance of his supreme and noble duty of contributing to the defense of the rights and honor of the nation, as the result of a war declared by the great representative body of the people, can be said to be the imposition of involuntary servitude in violation of the prohibitions of the Thirteenth Amendment, we are constrained to the conclusion that the contention to that effect is refuted by its mere statement.
The ruling in effect is reaffirming what everyone already believed. That is, that as a male citizen you have a reciprocal obligation to serve when called in return for your privileges as a citizen. Of course at the time your privilege was that at 21 you could vote. It wouldn't be until the passage of the 26th Amendment that all draftees would have the vote, but once again the argument was based on the concept of reciprocal obligation. But by then women's suffrage was already won, so the 26th applied to everyone. But Universal (18) suffrage was won because of the belief in reciprocal obligation. But to this day only men are obligated to serve.
In addition, failing to register for the draft can, in effect, disenfranchise you in many states as it is a felony.
The argument GWW is making is that this is the cultural norm, so ingrained it need not even be spoken, let alone codified in law thus: "men get the vote only if they are obliged to serve".
SCOTUS even provided a long list of other countries who did the same..
It is disingenuous to dismiss her claims on the basis that it wasn't so written.
It was the cultural norm and folks understood this as evidenced by some of the arguments against women's suffrage at the time. I however can not find sources showing that there was in fact opposition from women on that basis. (I'll let someone else bunk or debunk that claim).
So if you are a male citizen you do have an obligation that a female citizen does not have, and often the justification for that was his privilege which included voting. That it was not explicitly written in law that way does not invalidate the greater claims of entitlement and obligation. In 1918 SCOTUS did codify it into law, just not in so many words.
Additionally Karen Straughan often talks about Canada, being Canadian after all. So here is some evidence that around the time of Women's suffrage the idea that the right to vote was tied to the obligation to serve in the popular zeitgeist and so when war broke out was codified into law:
1917 - Wartime Elections Act - Gives voting rights to women with relatives fighting overseas. Voting rights are stripped from all "enemy aliens" (those born in enemy countries who arrived in Canada after 1902; see also Ukrainian Canadian internment).[44] Military Voters Act - gave the vote to all soldiers, even non-citizens, (with exception of Indian and Metis veterans)[45] and to females serving as nurses or clerks for the armed forces, but the votes are not for specific candidates but simply for or against the government.
I think it important to realize that GWW (et al) are trying to build a narrative showing that certain gendered privileges/entitlements/rights are tied to obligations, not necessarily in law, but in cultural norms.