When I delete an EKS/Kubernetes service and then create a new one - I believe the ingress hostname changes. The service is a load balancer - but we give at DNS name (CNAME) which points to the ingress hostname...so instead having to update Route 53 when we change the service, is there a way to avoid that?
So in short you can't avoid that. When you create a new EKS it will have to generate a new load balancer. Additionally, such NLB doesn't live in your account as the EKS is a managed service.
The cluster control plane is provisioned across multiple Availability Zones and fronted by an Elastic Load Balancing Network Load Balancer. Amazon EKS also provisions elastic network interfaces in your VPC subnets to provide connectivity from the control plane instances to the worker nodes.