Is there any metric tracking the time it takes until the first readiness probe is successful?

I saw the kubelet exposes probe_duration_seconds from version 1.25 prober package - k8s.io/kubernetes/pkg/kubelet/prober - Go Packages but I think this is tracking the duration in a per request level, but I’m interested in how much it takes for the pod from the moment the container is started until it answers k8s that is ready to receive traffic.

As a naive attempt to measure that meanwhile, I’m writing the result of this command date +%s to a file as part of the CMD statement in my Dockerfile and doing the diff against System/currentTimeMillis when the endpoint configured as readiness probe is called. But I would like to have something more trustworthy.