The biggest pain point is having to manually use cloudformation to create node pools. This is especially irritating when you just need to roll the Linux version on nodes -- takes half a day to do right. In GKE, it's just a button in the UI (or better, an easy-to-use API), and you can schedule maintenance windows for security updates (which are typically zero downtime anyway, assuming you have the right PodDisruptionBudgets). I think AWS fixed that. I remember when I used it, they said they had some new tool that would handle that, but you had to re-create the cluster from scratch. This was a couple years ago, and is probably decent nowadays.
There are other warts, like certain storage classes being unavailable by default (gp3), the whole ENI thing for Pod IPs, the supported version being way out of date, etc. EKS has always felt like "minimum viable product" to me -- they really want you to use their proprietary stuff like ECS/Fargate, CloudFormation, etc. If you're already on AWS and want Kubernetes, it's just what you need. If you could pick any cloud provider for mainly Kubernetes, it wouldn't be my first choice.
Having used EKS, GKE, and DOKS, I definitely prefer GKE. GKE is very feature-rich, and the API for managing clusters works well. The nodes are also cheaper than AWS. (I use DOKS for my personal stuff and I haven't had any problems, and it is free, but it's missing features like regional clusters that you probably want for things you make money off of.)
For what it's worth, there's an off-the-shelf terraform module for EKS that is far simpler to use than AWS' cloudformation tooling, which does allow you to pass in a custom AMI and multiple nodegroup configurations as input parameters.
EKS has actually come a long way in the last couple of years. They now have 'eksctl' that helps a lot with provisioning/de-provisioning the node pools. They're also pretty up to date with supporting new versions of k8s (1.21.2 now, I think). I definitely know what you mean about the 'minimum viable product' and their overall views on k8s though, as our account rep definitely tried to steer us away from EKS when we ran into problems with CronJob. They also don't seem to have any particular relationship with k8s development, so we ended up not able to use CronJob for more than six months while our issue (with k8s repo) just sat waiting for someone to look at it.
