But if you host your site on your infrastructure, and it goes down, you can't post status updates to tell people what's going on/ when you will be back online.
Its quite reasonable to not host your own homepage or mechanism of updating your customers IMO.
I disagree. Your website should run on your own infrastructure and a separate status page, under a different (sub)-domain should be operated from another AS (autonomous system) e.g. statuspage.io or whatever you like/prefer.
If you offer infrastructure services and don't tell people where and how you provide it, you can't be serious, too.