The app NewPipe and its web version https://invidio.us/ were recommended in the comments here. They allow you to use YouTube quite well without an account and without the official website/app. However, I imagine through loading videos, YouTube can still collect some data. (I might be wrong though, I don't know exactly how much they pipe through their servers.)

If you want to be completely private, you'll probably have to use TOR together with those alternative clients.