Multi-lingual 4K Video Downloader (4KVD) is frequently updated and features clear download links on the program's website; no ad traps here. The software does what it advertises in a simple interface: grabs videos up to 8K in quality and downloads to plenty of formats. Just copy a YouTube URL and click the Paste Link button to get started. 4KVD will even grab subtitles, entire playlists, and all the videos in a channel to which you subscribe. The sites supported are limited to the big names like YouTube, Vimeo, Facebook, and a few others, but covers most of what you need.
There are workarounds. One is using a free iOS file manager app like Documents by Readdle. Use the Safari browser to visit a YouTube video, and use the Share option to find Copy Link. Then go back to Documents, and use the built-in browser in the app to surf to a video download helper site like SaveFrom.net. Paste the link into the form (hold your finger down on it until you get "Paste" menu to pop up). The site will give you the links to download, and you can save the file to Documents. Hold down and drag the file up, until you're back on the main screen, then drag it to the Photos folder. You'll need to give Documents permission to access Photos the first time. You can then access the video like you would any video taken on the phone or tablet—in the Photos app.