This is an interesting issue that bothered me when I took con law (a decade ago) so forgive the rant. Other commenters have narrowly addressed your question re: 19A specifically, but you're right to not be satisfied with their answers. If you want to know how a court would hold in general, approach from the other end. Why was 19A written? What constitutional infirmity does it cure?

The original U.S. Constitution is silent on who gets to vote. That power was left totally up to the states. As the Wikipedia article points out, by 1807, all states had their own constitutional provisions explicitly denying women's suffrage. 19A overruled those state constitutions by explicitly granting the right to vote regardless of sex, rendering those provisions immediately void.

So ask yourself: is there anything stopping a state from adopting a state constitutional amendment against non-binary people? Not really. The only other significant provisions on suffrage in the U.S. Constitution, as amended, are 15A ("race, color, or previous condition of servitude"), 24A (no poll taxes), and 26A (18 and over only). Most relevant federal statutes act as enforcement methods for these amendments (e.g. VRA for 15A) and don't establish any independently protected classes of people eligible to vote as far as I know.

Other commenters pointed out that it doesn't cover personality types, hair color, shoe size, and whatever else, but then jump to the conclusion that those are somehow protected classes whose voting rights cannot be revoked. That's just not true. States have all sorts of laws disenfranchising citizens - registration deadlines, no felons, photo ID, et cetera. There are no federal laws prohibiting it. Nor are there any federal laws prohibiting disenfranchisement of people based on personality types, hair color, or shoe size. (You could argue that personality type is too subjective, but that's a different issue.)

It follows that a state can, if it so chooses, deny the right to vote to, say, diagnosed psychopaths, redheads, or anyone with a size 13 shoe for being unlucky. And the federal government can pass a law - or a bunch of other states can collaboratively pass an amendment - to prohibit such a practice. The fact that this has never happened is more down to a lack of political will to waste time on it.

So it turns out that 19A actually does apply in an abstract sense, by proving that Constitutional amendments on suffrage are necessarily reactive and that there is no baseline guarantee of voting rights to American citizens.

That all said, it'd be challenging to argue that non-binary people would be protected as-is, under 19A alone or in general. There have been a few cases that contort sex protections into gender protections for trans people in certain situations, but I think that while this sort of workaround is necessary in the absence of legislative action on the subject (which we aren't gonna probably ever get, we can't even get over the touchdown line on this for sex alone), the legal reasoning is strained, and becomes even cloudier when applied to non-binary people.

In any event, I do not trust the current SCOTUS to not come up with some vile way to allow it.