I want a Kubernetes StatefulSet with:
Is there a simple example on how to solve this typical requirement?
DaemonSets are exactly what you're looking for. They're a tool which will run a single Pod on every node for your resource. If a new node is created - a new Pod is created on it.
You can find further information onto how they're implemented at the following URL in the documentation:
https://kubernetes.io/docs/concepts/workloads/controllers/daemonset/