To show an icon on hover over a link on the browser side it would require browsers to make an HTTP HEAD request for each link to see if it has the "Content-Disposition: Attachment" header. That would add a lot of overhead to the page load, so I don't think it would be added to any browser.
However, it seems like a good candidate for a browser plugin for those who have fast connections and don't mind.
Note I said in case the download attribute is set on the link, in which case, no, it would not require any request, it just follows the instructions set in the HTML.
However, it seems like a good candidate for a browser plugin for those who have fast connections and don't mind.