You may already know but Cloudflare posted a blog on Sep 27, 2018 Introducing Cloudflare Registrar.
Cloudflare had many benefits like zero mark-up purcahses. This means that you pay the wholesale price that Cloudflare pays, this is unlike all (most?) other registrars that add their own fee on top of the ICANN fee.
The downside to this was that Cloudflare was only allowing transferring existing domains to their registrar, until today.
Looks like you can now register new domains through Cloudflare. This may be a staged roll out and it seems another Cloudflare account didn't have access to this button.
Here is what the searching looks like!
Oh and if you are interested in getting a status page for your website checkout my other dev.to post, I built a status page using React and GitHub Issues