Make PostPicker compatible with post type-specific search endpoint

### Description

Currently, the `PostPicker` uses the core search endpoint with `type: post`. I would like to be able to have the `PostPicker` component search for posts within a specific post type by using that type's REST endpoint and the `search` parameter.

### Use Case

It's possible for a post type to be visible in the REST API but not `public`. These post types are excluded from the core REST API search endpoint, which includes only post types with `{ public: true, show_in_rest: true }`, which means they aren't visible to the `PostPicker`. I would like to be able to use this component to search these types.