Indeed, the networking layer that EKS forces upon you is quite strange. For example, you can only run 17 pods on a t3.medium instance because of how IP addresses are managed in your VPC (and because of various mandatory daemonsets that run pods on every machine, like aws-node). Most people will likely find out about this very low limit not in normal operations, but when one AZ blows up and pods are rescheduled on other AZs. You will hit the pod limit, things will be down and nothing new will schedule. And Amazon provides no meaningful monitoring for health at the control plane level; you need to bring all that yourself.
Managing worker nodes is also a huge pain. They recently released a tool that manages some of it, but it doesn't work with older clusters. To do something simple like fix a vulnerability in the linux kernel, you have to make a new CloudFormation stack (involving cutting and pasting a ton of random stuff; the template ID, the AMI ID, etc.), edit several security groups, start the new nodes, make sure they join, drain the old nodes, delete the old stack, edit security groups. Upgrading the Kubernetes version in the cluster is a similar story, especially if you skipped an intermediate version. (That said, incidental changes like adding more nodes is easy. It's a lot rarer than upgrading Linux or the k8s version, however.)
Their load balancers other than Classic don't work every well either. I would prefer to use a NLB for all incoming traffic (terminated by Envoy in the cluster which does the complicated routing) but that apparently results in kube-proxy not working right... and every time you change a set of pods whose selector matches the NLB service, it changes all the IP addresses in DNS, and traffic just stops arriving until the DNS TTL changes. It is very broken and the docs should say "never even think about using this" instead of "it's beta and here are some weird caveats that probably won't dissuade you from using it".
Anyway, my overall impression of EKS is that Jeff Bezos walked into someone's office, said "you're all fired if we don't have some half-ass Kubernetes service in the next 3 months" and walked out. The result is EKS. It's wayyyyyy better than being locked into Amazon's crazy stuff (CloudFormation, etc.) but it's not as good as what other managed k8s providers offer.
If I were to setup k8s again, I would just buy my own servers and self-host everything. Dealing with Things That Can Go Wrong With Servers is much easier than dealing with Things That Can Go Wrong With AWS. (But I already have a datacenter, can get the machines a 100Gbps connection to the Internet for free, and have a team of network engineers to tell me which CNI to use. If I didn't have that I would just use GCP.)