GitHub Actions: Allow writing PRs when auto-cherry-picking

[t-hamano]

• Follow-up Persist credentials when cherry-picking commits to a release branch #72556
• See the discussion on Slack: https://wordpress.slack.com/archives/C02QB2JS7/p1761133096282149?thread_ts=1761097846.014379&cid=C02QB2JS7

What?

When a PR is tagged with the Backport to WP X.X Beta/RC label, GitHub Action triggers it to automatically cherry-pick the PR into the wp/x.x branch. Some issues were resolved in #72556, and the cherry-pick itself now works correctly, but the following two workflows don't work:

• Remove the Backport to WP X.X Beta/RC label.
• Comment about the cherry-pick results on the PR.

My guess is that the reason is that the permission was limited to read by #69126.

https://github.com/WordPress/gutenberg/pull/69126/files#diff-712273121c4cf0e79eab65a0cbc9ee462c9dfef1ffc2a72f164bae6b46e3d625R29

Testing Instructions

This PR is difficult to test, but let's see if the workflow works correctly after this PR is merged.

[github-actions]

The following accounts have interacted with this PR and/or linked issues. I will continue to update these lists as activity occurs. You can also manually ask me to refresh this list by adding the props-bot label.

If you're merging code through a pull request on GitHub, copy and paste the following into the bottom of the merge commit message.

Co-authored-by: t-hamano <wildworks@git.wordpress.org>
Co-authored-by: ellatrix <ellatrix@git.wordpress.org>
Co-authored-by: Mamaduka <mamaduka@git.wordpress.org>

To understand the WordPress project's expectations around crediting contributors, please review the Contributor Attribution page in the Core Handbook.

[ellatrix]

It's quite difficult to test workflows. Happy to try this.

[Mamaduka]

Let's give it a try. It's an easy change to revert.

[github-actions]

Flaky tests detected in 970ee74.
Some tests passed with failed attempts. The failures may not be related to this commit but are still reported for visibility. See the documentation for more information.

🔍 Workflow run URL: https://github.com/WordPress/gutenberg/actions/runs/18715490248
📝 Reported issues:

• [Flaky Test] Should insert content using the global inserter #57525 in /test/e2e/specs/widgets/editing-widgets.spec.js
• [Flaky Test] ensures List View global shortcut works properly #51533 in /test/e2e/specs/editor/various/list-view.spec.js

[t-hamano]

Auto cherry-picking seems to work correctly now 🚀

#72565 (comment)