is the SUSE Private registy a part of the SUSE Rancher support subscription offering? https://www.suse.com/c/suse-private-registry-harbor/

No, it's outside the scope.

thank you @tall-school-18125 for the prompt answer! and there is no other private registry included in the Rancher support offering, is it?

Well, Rancher requires some kind of registry in order to work, either Dockerhub or a private registry for air gap environments. So if you got Rancher support, it would cover any issues with troubleshooting Rancher, which could involve problems with pulling images, but not any bugs in the private registry itself

makes sense, thank you!