Docker compose has always been great for running some containers on a local machine, but I’ve never found it to be great for deployments with lots of physical nodes. k8s is certainly complex, but the complexity really pays off for larger deployments IMO.
I do this for multiple reasons, one is that I find it easier to use Kubernetes as the backend for Patroni, rather than running/securing/maintaining just another etcd cluster. But I also do it for observability, it's much nicer to be able to pull all the metrics and logs from all the components. Sure, it's possible to set that up without Kubernetes, but why if I can have the logs delivered just one way. Plus, I prefer how self-documenting the whole thing is. No one likes YAML manifests, but they are essentially running documentation that can't get out of sync.
I will say that even though the StatefulSet pins the pod to a node, it still has advantages. The StatefulSet can be scaled to N nodes, and if one goes down, failover is automatic. Then you have a choice as an admin to either recover the node, or just delete the pod and let the operator recreate it on some other node. When it gets recreated, it resyncs from the new primary and becomes a replica and you’re back to full health, it’s all pretty easy IMO.
With Docker Compose, the abstraction level you're dealing with is containers, which means in this case you're saying "run the postgres image and mount the given config and the given data directory". When running the service, you need to know how to operate the software within the container.
Kubernetes at its heart is an extensible API Server, which allows so called "operators" to create custom resources and react to them. In the given case, this means that a postgres operator defines for example a PostgresDatabaseCluster resource, and then contains control loops to turn these resources into actual running containers. That way, you don't necessarily need to know how postgres is configured and that it requires a data directory mount. Instead, you create a resource that says "give me a postgres 15 database with two instances for HA fail-over", and the operator then goes to work and manages the underlying containers and volumes.
Essentially operators in kubernetes allow you to manage these services at a much higher level.
As someone who has operated Postgres clusters for over a decade before k8s was even a thing, I fully recommend just using a Postgres operator like this one and moving on. The out of box config is sane, it’s easy to override things, and failover/etc has been working flawlessly for years. It’s just the right line between total DIY and the simplicity of having a hosted solution. Postgres is solved, next problem.