His argument is that "the button" has no depth and therefore is not perceived as a button.

My argument is that the bright color makes it stand out without using depth (or rather, using depth in another dimension than Z), and that it's the checkmark icon that might make it feel non-interactive (checkmarks feel more like feedback than action, even on checkboxes, where it's the box itself that makes it register as actionable). A "login" or "proceed" label would have worked, or as an icon, something giving sense of proceeding further, like a play or fast forward button would have worked better in conveying the sense of action. Still, the structure of the UI makes it clear that this is a two-field form and that the third item means "go ahead", which creates a cognitive dissonance with the "feedback" feeling of the checkmark.