They don't need to. Chrome does extremely well on Windows without "owning the OS".
> nor the advantage of advertising on the google home page like chrome
Very few people go to that page anyway.
> The "customers" they are courting are security/privacy zealots who constantly want them "just work on the browser" but never seem to have an answer on how to pay for that.
They already get half a billion dollars a year from Google for very little in exchange. They just need to spend that money on the website browser and not nonsense.