The status page is static, in that there is no server-side rendering/processing other than just serving static files. You need only host the resources for the status page, but the files with the check results are typically stored somewhere else like S3. The status page downloads those check files with JavaScript and renders the results client-side. Hope that answers your question!
Why would you default to S3 storage? You create a stand-alone product for self-hosting, complain about cloud providers going down, but then use a cloud provider for your collection?
You can implement and plug in any storage provider, including local storage if you want. The point is you don't have to rely on a proprietary, locked down status page service.
I'm the author of Checkup; hope others find it useful!