I'm actually on a bridge call with Google Cloud, we're a large customer -- I just learned today that their status page is not automated, instead someone actually manually updates it!
That's the case with every status page. These pages are managed by business people not engineers, because their primary purpose is to show customers that the company is meeting contractually defied SLAs.
Maybe or maybe not, but someone with nothing better to do than monitor that page out of boredom might “get on the horn” with lots of people to complain if a green check mark turns to a red X.
They aren't automatically based on that page, but seeing a red status makes it too easy for customers to point to it and go "see you were down, give us a refund".
The best way to consistently having good "time to response" metrics, is to be the one deciding when an incident "actually" started happening, if at all :)