You would want to host the status page on completely independent infra, such that if your service goes down, it doesn't take down the status page as well. This

You would want to host the status page on completely independent infra, such that if your service goes down, it doesn't take down the status page as well. This includes DNS, hence the separate domain. For example, with Facebook's recent outage, because they hosted their status page on the same DNS server as their regular service, they weren't able to communicate the outage on their status page.