I built something similar to this when I was first learning to program. Except mine lets you perform a one-time search for videos containing keywords, similar to how you would normally search YouTube. So there are no notifications or anything.
I don’t know if it even works anymore and I’m sure the code is atrocious. But I remember that I would just scrape YouTube pages for video IDs, and then use an API that returned video captions for a given ID [1]. I could see how OP would do something similar.