Why does this button exist? Is it like a gamut warning? It only appears in the gradient color picker

The out of gamut warning appears to be hard-wired to your working CMYK, not the active Proof conditions.

In other words, mostly useless for any practical purpose. Just ignore it.

The little cube indicates the color in not web safe. Clicking on the square forces the color to the next web safe one. Just the thing for Netscape Navigator 🙂

You may also see a triangle, which does indicate out of gamut, and the square by this will force the color to be in gamut.