In my case, the step that was missing was to add the website to my list of trusted sites in Internet Explorer. That takes the list of pre-requisites to:
Enable the WebClient service in Windows (if you're using a Windows Server environment, you need to enable the
Desktop Experience
feature of Windows Server)Add the website to the Trusted Sites list in Internet Explorer
Make sure that you're using a 32 bit version of Internet Explorer - Windows Explorer mode isn't supported in the 64 bit versions of the browserWindows Explorer mode isn't supported in the 64 bit versions of the browser.