You can find public instances at https://github.com/omarroth/invidious/wiki/Invidious-Instances. Not all of them have the download button shown. Though, if in firefox, right click into the video field and you should be able to copy the link to the video file itself (which should be something like https://r6--5s--d-f7.googlevideo.com/.....- and a long URL). You could then download it with
`curl --output videofile.mp4 <url>`
I haven't used mpv yet for that purpose, so I don't know. But maybe @amolith can help you out with that.