Unless I'm mistaken, it's currently not possible to set up fargate-only clusters with the quickstart.
The coredns deployment is not patched to not require ec2 compute type (cf. https://docs.aws.amazon.com/eks/latest/userguide/fargate-getting-started.html).
The aws-load-balancer-controller deployment is not compatible with fargate (missing vpc-id and region options). Cf. point 5.d at https://docs.aws.amazon.com/eks/latest/userguide/aws-load-balancer-controller.html.
Error taken from one of the aws-load-balancer-controller pods:
{"level":"error","ts":1625053562.5521672,"logger":"setup","msg":"unable to initialize AWS cloud","error":"failed to introspect vpcID from EC2Metadata, specify --aws-vpc-id