You can inspect the controls of the native Chrome Video Player by activating the shadow DOM in Settings|Preferences -> Elements -> Show user agent shadow DOM
After that you can inspect the players buttons.
Now the problem is that the download button cannot be accessed via CSS for some reason.
video::-internal-media-controls-download-button {
display:none;
}
won't work. Even selecting the preceding button and targeting its neighbor using + or ~ won't work.
The only way we found yet was nudging the button out of the viewable area by giving the control panel a greater width and making the enclosure overflow: hidden
video::-webkit-media-controls {
overflow: hidden !important
}
video::-webkit-media-controls-enclosure {
width: calc(100% + 32px);
margin-left: auto;
}