Automatic merge from submit-queue. If you want to cherry-pick this change to another branch, please follow the instructions <a href="https://github.com/kubernetes/community/blob/master/contributors/devel/cherry-picks.md">here</a>.

[e2e service] Move apiserver restart validation logic into util

**What this PR does / why we need it**:
Follow up of #60906, on GKE apiserver pod is invisible on k8s, hence test is failing.

This PR bakes the restart validation logic into the util function instead so it could be env-awared.

**Which issue(s) this PR fixes** *(optional, in `fixes #<issue number>(, fixes #<issue_number>, ...)` format, will close the issue(s) when PR gets merged)*:
Fixes #60761

**Special notes for your reviewer**:
Sorry for the noise.
/assign @rramkumar1 @bowei 
cc @krzyzacy 

**Release note**:

```release-note
NONE
```