Wasn't there that time AWS had some outage, but the red-circle picture (for failure) was itself hosted on the unavailable AWS service, so their status page didn't show it as being unavailable? Or am I mis-remembering something/repeating an urban legend?

Nah, you're remembering correctly, this happened. The failure icon was hosted on S3, but S3 was down. It was a great moment.

Here's their tweet about it: https://twitter.com/awscloud/status/836656664635846656

Thank you for confirming, and the citation!

Yep, the icons were hosted on S3, which was unavailable

https://www.theregister.co.uk/2017/03/01/aws_s3_outage/

Yes, that was the S3 outage last year...

https://news.ycombinator.com/item?id=13756111

These pages at Google/Amazon/Netflix are updated by the incident response team, so it's a manual task.

Edited my comment to just say the same thing. Haha What's the point of them then?

It's certainly helpful to know that Google is aware of the problem and fixing it.

I guess technically the "server" is responding, so...