Ever since the the 7.x update which made the Videos section way less usable (like scrolling up and down for no reason or the removal of the list view) I also noticed that tapping "download" seems to do nothing.
In older versions, when you tapped download, the icon changed. I can't remember what exactly it look like (I think it was a circle showing the download progress, or a different icon to indicate that downloading was queued) but in 7.x, nothing changes. If you close out of the video, and out of the videos section and go to the downloads section, it does show the download progress there. But it's really confusing in the videos section when there is no indication if the download button was pressed or not.