Init container livenessProbe



I noticed an issue that occurs occasionally, the init container is stuck and with that the pod is stuck. A manual pod restart solves the issue.
Going over the k8s docs I found:

Use activeDeadlineSeconds on the Pod and livenessProbe on the Container to prevent Init Containers from failing forever. The active deadline includes Init Containers.
Init Containers - Kubernetes

Not sure I fully understand how the container livenessProbe could help here. If the init container did finish loading then the container can’t start. When should the livenessProbe kick in at such a case?
I understand that the activeDeadlineSeconds will restart the pod but I don’t want to limit the container runtime, only the init container.
Why isn’t it possible to configure a livenessProbe for the init container?