Google Chrome is now shipping with a download button for videos that are just embedded videos (i.e. not MSE):
Yes, this is possible now, at least at the time of writing, you can use the controlsList attribute:
It seems this was introduced in Chrome 58, and the documentation for it is found here: https://developers.google.com/web/updates/2017/03/chrome-58-media-updates#controlslist
Developers can now customize media controls such as the download, fullscreen and remoteplayback buttons. Usage in HTML:
There is even an official sample page: https://googlechrome.github.io/samples/media/controlslist.html