Indeed, and that's exactly what I'm using on desktop. But on mobile, I prefer the app, and that one doesn't keep a local history.
My suggestion is to stop worrying about hypothetical privacy problems and just use the watch history feature. Google will even automatically clear old history for you if you want.
HN nerds get the worst user experiences because they have these random lines in the sand. I’ve been guilty and often try to break out of it when I realize that the people who don’t know better are enjoying themselves more.