Update update-schedule workflow have been failing

**This is a Bug Report**

The update-schedule workflow has been silently failing or run without creating it's intended PR. All schedule updates since then have been manual.

Root cause: The stale update-schedule branch has diverged and branch protection (EasyCLA required, no force-push) prevents the action from pushing to it. The action fails.

remote: error: GH006: Protected branch update failed for refs/heads/update-schedule.
remote: - Required status check "EasyCLA" is expected.
remote: - Cannot force-push to this branch

Ref: https://github.com/kubernetes/website/actions/workflows/update-schedule.yml

Start with the update-schedule workflow at .github/workflows/update-schedule.yml and review its branch and push behavior alongside the reported GH006 errors. Verify the workflow can update the schedule branch despite EasyCLA and branch-protection requirements, and confirm that a scheduled run creates its intended pull request without silently failing.

Written by the indexing model from the issue text.