They should have pro-active monitoring system and update status page automatically when it detected disruption. Especially when you account to it's extremely hard to get human supports from Google because they automated it.
I think I'd prefer a status page that reflected that, or even no status page at all with an explanation, rather than a lying one, though.
The on-caller for the specific service decides on how to produce.
The status dashboard is something which will be (manually, yes by hand) updated by an operations employee, who is a couple of layers behind the actual SWE/SRE who gets the page.