I have no programmatic access for the user who created the cluster only graphical. I tried a lot of things and I always get access denied on kubectl as I have to use another IAM identity that has programmatic access. I assume there is some trick with STS but I don't have the access keys of the root account. I guess if I give graphic access to the IAM identity that I want to use, it will work out, but I want to know if there is a way with my current situation. Also using the aws-cli is not an option for creation in the given scenario. The admin who created my user won't allow my federated IAM to have access keys.
As said in the question summary, the issue is that I have created an EKS cluster from IAM user that doesn't have programmatic access. I have tried what I have suggested as scenario that I would like to avoid. Only solution in this case was to create another IAM identity that will create the cluster and give it programmatic access. As I am a federated user from OKTA, I have admin rights but using the console on the created identity is a security bridge, so I have allowed it only to login, create the cluster and then disabled the console access. The from the newly created identity I could use kubectl to control the cluster and add other IAM identities from the auth configmap.