I've encountered one scenario where element queries would be incredibly useful: web app designs that include sidebar navigation that toggles open and collapses. This design pattern makes responsive design a little bit more difficult. The main challenge is that the width of the main content pane changes depending on the state of the sidebar. Your media query to scale down an element or rearrange a layout, often has to default to happening at a greater window width than otherwise necessary (if the sidebar is collapsed), because of the potential for the sidebar being open. There are some gross ways to work around this with JS, but I don't think it's worth it. An element query would make it easy to watch the width of the main content pane, independent of the sidebar pane's state.
If you create an account at Stipple.com and log in, you'll see an example of what I'm talking about (disclosure: I'm an engineer at Stipple).
If you create an account at Stipple.com and log in, you'll see an example of what I'm talking about (disclosure: I'm an engineer at Stipple).