25798: Improves reliability of docs rebuild pipeline

[apbassett]

No description provided.

[howso-mario]

Reviewed by Hermes Agent. Verdict: comment / no blocking findings.

Evidence inspected: PR diff and repo guidance; status rollup for run 30305742261 (all build/smoke docs/final-check lanes green, release-only lanes skipped as expected); final-check log shows check-docs-commit, set-pr-version, build, and rebuild-docs all success; rebuild-docs log downloaded the ARM64 artifact and found no generated-doc changes; local git diff --check origin/main...HEAD was clean.

The final-check rewrite looks materially safer than the old required-check wrapper because it runs under always(), includes set-pr-version/build/rebuild-docs in needs, and requires exact success instead of treating skipped dependencies as acceptable. I left one inline non-blocking reliability hardening note for the docs-bot branch-movement window. No merge action taken.

[howso-mario]

Approved at head f456598. Verified the docs rebuild workflow records the checked-out SHA, fetches and compares the live PR branch head before push, and exits successfully without pushing if the branch moved, addressing the prior reliability note. Inspected current diff and review thread inventory; no blocking findings remain. CI run 30369245810 is complete: required build, cross-platform smoke, rebuild-docs, and final-check lanes passed; release-only lanes were skipped as expected.