Ensure page/post stubs are available in available nav menu item searches

[westonruter]

The ability to create new post/page stubs was added in #34923. However, these posts/pages do not appear when doing a search. Similarly, if new posts/pages are created during a customizer session (such as via the Customize Posts plugin) then similarly these posts will not appear. The reason stems from the fact that the customized state is not being included in the admin-ajax requests and the settings are not previewed. Something similar was done for update-widget requests in [33488] for #32814.

The required code can be seen in Customize Posts, which has a polyfill: ​https://github.com/xwp/wp-customize-posts/pull/288/commits/de381a06e527311182e311270f39408462251563

[westonruter]

Also need to make sure that the nav menu items created from starter content appear among the available nav menu items. This is closely related to code touched by #36697. The Ajax request to load items needs to include the customized state.

[westonruter]

In 39138:

Customize: Ensure page/post stubs are included in listings and searches for available nav menu items.

Include the customized state in the Ajax requests to load items and search items.

See #38573.
Fixes #38122.