I don't know of any website but yt-dlp is great for downloading videos.

They already suggested yt-dlp, but also have a look at JDownloader. You can paste a webpage url and it will fetch all the elements it can download, video included. You can also create download collections in order to download the same files again (or in another computer with JDownloader installed).

If the only thing you are missing is youtube download ability, why not use one of the many options to do that from a handset? Newpipe or Spottube are on fdroid. https://invidious.io/ allows you to do it from a browser.

On Android, it's a bit complicated but you if you are experienced you can use termux and yt-dlp.

You can use yt-dlp from a desktop too.

For youtube on android you can use newpipe. It allows streaming as well as downloading.