[DOCS-XXXXX] Add Set Up a Standalone Private Action Runner

[irving-santiago]

What does this PR do? What is the motivation?

Fixes DOCS-XXXXX

Adds a new page for setting up a standalone private action runner (UI-driven install, programmatic install, connecting, managing, and updating the runner), split out from the Datadog Agent setup path added in a previous slice in this stack.

Also retires the standalone "Update the Private Action Runner" page: its content is now fully covered by the "Update the runner" section on this page (standalone) and on "Set Up in the Datadog Agent" (Agent-based, added in the previous slice), so the old page is redundant. Removes it from the nav and adds a redirect alias pointing to "Set Up in the Datadog Agent" (the more common, recommended path) for the old URL.

Merge readiness

• Ready for merge

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AI assistance

Claude Code carried this page's content over from a pre-drafted, QA-verified draft and ran Vale. Fixed a real bug in the draft: several GCR image-tag references were missing the v prefix used elsewhere in this repo for the same image. Also authored the "Update the Private Action Runner" retirement (page removal, nav update, alias) directly against the repo, at the user's request after a post-implementation structure review against the original plan.

Additional notes

[irving-santiago]

Warning

This pull request is not mergeable via GitHub because a downstack PR is open. Once all requirements are satisfied, merge this PR as a stack on Graphite.
Learn more

• [DOCS-XXXXX] Add Get Started with Private Actions (Execution Policies quickstart) #39054
• [DOCS-XXXXX] Add Private Actions Reference #39053
• [DOCS-XXXXX] Rewrite Run a Script for Execution Policies #39052
• [DOCS-XXXXX] Add Execution Policies #39051
• [DOCS-XXXXX] Add Authorize Private Actions #39050
• [DOCS-XXXXX] Add Enrollment and Ownership #39049
• [DOCS-XXXXX] Add Set Up a Standalone Private Action Runner #39048 👈 (View in Graphite)
• [DOCS-XXXXX] Rewrite Set Up in the Datadog Agent, add Execution Policies enrollment #39047
• [DOCS-XXXXX] Move Handling Private Action Credentials under Connections #39046
• [DOCS-XXXXX] Rewrite Private Actions Overview for Execution Policies #39045
• Add tracking doc for the Private Actions / Execution Policies stack #39044
• master

This stack of pull requests is managed by Graphite. Learn more about stacking.

[github-actions]

Preview links (active after the build_preview check completes)

New or renamed files

• https://docs-staging.datadoghq.com/irving.santiago/par-ep-04-set-up-standalone/actions/private_actions/set_up_standalone

Removed or renamed files (these should redirect)

• https://docs-staging.datadoghq.com/irving.santiago/par-ep-04-set-up-standalone/actions/private_actions/update_private_action_runner

Modified Files

• https://docs-staging.datadoghq.com/irving.santiago/par-ep-04-set-up-standalone/actions/private_actions/use_private_actions

[datadog-prod-us1-5]

⚠️ Warnings

🚦 1 Pipeline job failed

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