The complexity difference between bog-standard ECS+Fargate and EKS+Fargate deployments rounds down pretty small. Biggest I've seen: ALB integration, IAM integration, and maybe certificate management. Most of that stuff is out-of-box on ECS, but on EKS you need some extra containers or configuration to watch the K8s API and provision stuff for you (if you want to use it; you can also just go pure-k8s) (edit: just to be clear, they provide all this for you; its not out-of-box, but its easy-to-add-box e.g. [1]).

An argument could be made for something like CodeDeploy being better integrated on ECS, but that's more of a "k8s doesn't need CodeDeploy but ECS might" kind of thing. And even then, I wouldn't touch it.

An argument could also be made that upgrading ECS clusters is a bit easier, as the cluster itself, uh, doesn't have a "version". But on Fargate, its pretty painless on EKS, and Fargate ECS tasks do have a "platform version" that generally doesn't have to be worried about (version: LATEST), but is nonzero nonetheless.

Which is really to say that both ECS and EKS puke complexity, because its AWS, but the volume is pretty similar.

[1] https://docs.aws.amazon.com/eks/latest/userguide/aws-load-ba...