The staging suffix needs to have its own DNS zone since Let's Encrypt will look for the presence or absence of a CAA record. If this fails they will refuse to issue an SSL for the subdomain. If the nameservers are delegated to your Enhance cluster, adding it as a website is the easiest way. If it's using external DNS, you should create the zone there.
Automatic SSLs are processed in a queue and only when your server detects that the DNS is pointed correctly. It can be that your server's DNS resolvers cache an NXDOMAIN response before the DNS is ready. Requesting the SSL manually via the "security" tab overrides this check and allows the SSL to be issued instantly.