Not updating that status page when the core domain goes down: less good
At that point it's worse than what you already know from your browser - it may show the service is having issues when you can access it, or that the service is ok when you can't.
Worst case you have more data points to draw conclusions from. Status page red, works for me? Hmm, maybe that's why the engineers in the other office are goofing off on Slack. Status page green, I get HTTP 500s? Guess I can't do this thing but maybe other parts of the app still work?