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.
Another workaround: Go back to the desktop and try AnyTrans ($39.99 for a single computer), a file manager for iOS devices that has an integrated downloader supporting 900 sites, including YouTube and Facebook. It'll transfer the videos to the iPhone for you over the USB cable. Even if you don't pay for AnyTrans, the download option remains and is free forever.