A click handler can be doing a lot of things that aren't much like a button, like letting you close a modal if you click outside of it, capturing mouse events for a game, or passively recording events for analytics. All that a click handler tells you is that there's some code that sometimes cares about some clicks somewhere inside that element.

All of those seem like examples of things you’d want your screen reader to tell you about.

Sure.

What is the screen reader's plan for determining the purpose of the attached JavaScript?

Same as a button. Read the text in the element. If no text, skip it or say "clickable".

I'm not arguing a div is better. I'm just saying screen readers could announce a div with an onclick if they wanted to.

Also a click handler on a div isn't going to do much for someone who isn't using a mouse, which would include a lot of screen reader users.

I can confirm. I got decent with VoiceOver as the "guy on the team who gives a shit about this" (unofficial title) and yeah, it stinks. Also, you don't necessarily move through the page linearly with a screen reader; the rotor lets you jump between links, or jump to forms, or jump to navs. The more semantic your markup, the better it will be populated.

It's worth noting that VoiceOver is well-known for being a lot more lax than most other screen readers. Other options tend to be a lot more strict around standards and what they'll announce.