One of the things that drives me crazy about YouTube is that if a video gets taken down, it shows up as a "This video is no longer available" with no further metadata. I am far, far more uptight about no knowing which video was removed than I am about the fact that it is no longer available. I have put serious thought into creating a tool that would automatically yt-dlp every video I open to a giant hard drive and ap…
In the end, I decided it is not worth it. In the scenario you described, I would take the video link/ID and paste it into Bing and Yandex. There is large chance they still have that page cached in their index.
FWIW if you are going to create your own tool, my advice will be to make it a browser extension, and try to pull the video straight from YouTube's element.