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.
If you’re looking to download a whole YouTube channel at one time, this is definitely one way that you can do it. WinX YouTube Downloader can download over 1000 URLs. Even more, it has support for different YouTube playlists and channels and even gives you the ability to record your own live videos, so this is one platform that can definitely achieve a whole lot more than you might have thought. On top of that, it doesn’t give you any kind of ads (because who wants to watch those?) and it lets you download from just about any kind of platform and into just about any kind of file.