I have a domain that can only be registered as a 3rd level from the TLD (.name gTLD).
If I try to add the 3rd level CF says “Please ensure you are providing the root domain and not any subdomains (e.g., example.com, not subdomain.example.com)”
From what I know about that TLD, there’s a company that’s registered a ton of .name domains, then sub-registers the subdomains in their own ecosystem. So those subdomains won’t work with Cloudflare.
Since you haven’t posted the subdomain, this is just a theory. Where did you register your third level subdomain?
In this case, IIRC from years ago I believe the .name gTLD owner Verisign had reserved the 2nd level domain (probably because it’s a common surname) and only allowed 3rd level registrations.
I don’t believe it’s a separate company because I was able to get a special setup for email forwarding at the registrar which relays through mx.nic.name. Ie, <3rd level>@<2nd level>.name forwards to @<3rd level registered domain>.name.
IIRC I think I first registered with Dyn.com and where the registration is currently. They also assisted with enabling the email forwarding at the register.