You know, a lot of this depends on someone clicking a link. If the organization's mail system turned every link into a non-clicky, People would have to manually view the link's source, select it, and copy-paste into a browser's address bar. I'd expect that many people would not bother, so sending links by e-mail would diminish quickly.
Of course, orgs would have to stop sending "time to change your password" links, but that should have ended years ago.