It's an interesting idea!

I don't think the button experiment gives the right results, because some people with "gender aphorhia" (i.e. "I really don't care that much about my gender") would score positive, yet have experiences and outlooks that are are far less similar to a gender dysphoric trans person than to a gender euphoric cis person (because whether you love it or whether it's whatever, as long as you're not clashing with expectation, everything is fine).

But I think the concept potentially has some merit.

However, let's explore the idea a little bit more. If there's a difference between the subconscious sex idea and the "a woman is someone who says they're a woman" idea (which I will, hereafter, call the "gender declaration" idea), there should be some scenario where we can tell the two apart.

Because the key feature of gender declaration is the declaration, you can't really tell the two apart unless it is possible for the declaration to be wrong.

What would a wrong declaration look like? It can't just be that they're lying--you can induce people to lie about almost anything given severe enough consequences (and our society can deliver pretty unpleasant consequences for expressing anything but embrace of gender). They have to actually be mistaken.

So what would it look like? How would you tell? Someone who is assigned D at birth says they're E, but they're wrong: they're D. And, someone who is assigned D at birth says they're D, but they're wrong: they're E. How, even in principle, could we know? (Where D and E are male and female, or vice versa.)

Feel free to use scenarios that aren't actually possible, like the sex-swap magic used with the button example.

The bottom line, though, is that if there is no distinction between gender declaration and subconscious sex, we may as well just stick with gender declaration because it's clearer how to make that actionable.