Expose privacy policy page through REST API

[oandregal]

Related ​https://github.com/WordPress/gutenberg/pull/82422
Pull request ​https://github.com/WordPress/wordpress-develop/pull/13399

Backport of ​https://github.com/WordPress/gutenberg/pull/82422.
Trac ticket https://core.trac.wordpress.org/ticket/66045#ticket

The page assigned in Settings > Privacy is stored in the wp_page_for_privacy_policy option, but the option is not registered with show_in_rest, so the /wp/v2/settings endpoint cannot report it. This registers it as page_for_privacy_policy, alongside page_on_front and page_for_posts, so the editor can show a "Privacy Policy Page" badge next to that page, as the classic pages list does.

Changes:

• register_initial_settings(): register wp_page_for_privacy_policy with the REST name page_for_privacy_policy.
• rest_restrict_privacy_policy_page_setting_update(): ignore updates from users without manage_privacy_options. The settings endpoint only checks manage_options, and on multisite manage_privacy_options maps to manage_network, so a site administrator could otherwise change the page over REST but not in Settings > Privacy.
• Settings controller tests: expect the new key, and cover updating the setting with and without the capability.
• REST fixture: add the new setting to the schema and the settings response.

Trac ticket:

## Use of AI Tools

AI assistance: Yes
Tool(s): Claude Code
Model(s): Claude Fable 5.1
Used for: Drafting the change and the tests; reviewed and tested by me.

Let's also include (if needed) ​this change in tests for multisite.

@ntsekouras ported at ​https://github.com/WordPress/wordpress-develop/pull/13399/changes/599f64378d62cf4eaac21ea520de5d62947f27ee

[oandregal]

In 63526:

REST API: expose privacy policy page in settings endpoint.

Props oandregal, ntsekouras, joen.
Fixes #66045.