Eric has been writing about tech for over 27 years. He was on the founding staff of Windows Sources, FamilyPC, and Access Internet Magazine (all defunct, and it's not his fault). He's the author of two novels, BETA TEST ("an unusually lighthearted apocalyptic tale"--Publishers' Weekly) and KALI: THE GHOSTING OF SEPULCHER BAY. He works from his home office in Ithaca, New York.
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.
Refresh the page and try again. Sometimes it is better when no other applications or software are running. For example, close Word, Excel, etc., and/or anything else that you are not using at the moment, and make sure you are not connected to Skype, Facebook or your email while doing the video download so your computer is not "busy" running heavy apps or software.