Google Chrome is now shipping with a download button for videos that are just embedded videos (i.e. not MSE):
Demmongonis solution does work but be aware it can lead to unwanted results.
Android/Chrome sometimes, depends in the video I guess and other factors, adds buttons at the right of the download-button. i.e. the casting-button (there is no way to select it). It will make the download-button to remain visible and the last button to get hidden (casting-button)
Update
It is posible now to hide the download button using the controlsList attribute: