I love this. Could you please consider changing the example from a clickable div – which is an accessibility nightmare – to a button?
This removes all the preset browser css styles. An accessible,styled <button> is much easier to achieve than an accessible, styled <div>
I suspect the answer is that screenreaders don't look for all clickable elements and are naively focusing only on buttons.
Logically it should be the reverse. Screenreaders can learn how to detect all clickable elements instead of shoehorning a form control element into places.