WP_Query on different site in a multisite setup

I am trying to retrieve search results from one multi-site site (“Staffsite” – id: 2) and display them alongside the search results on another site that’s part of the multi-site setup (“Flagship” – id: 1). I currently have the following code in search.php on Flagship:

$args = array(
    'post_type' => 'staff',
    's'         => get_search_query(),
);

switch_to_blog( 2 ); // switch to Staffsite

$staffsite_query = new WP_Query( $args );

var_dump( $staffsite_query->post_count ); // 0

// Do something with post data

restore_current_blog(); // return to Flagship

The query works as expected when used in a template on Staffsite (replacing get_search_query() with an actual value), returning all the relevant posts, but does not work when used from a template on Flagship in combination with switch_to_blog().

For what reason does this not work?

Alternative option could be using the Rest API:

Example (post):

Search for Jane Doe in posts:

https://example.com/staffsite/wp-json/wp/v2/posts/?search=jane+doe

Example (custom post types):

If the public and searchable custom post type (e.g. staff) has show_in_rest set to true then we can search for Jane Doe in staff posts:

https://example.com/staffsite/wp-json/wp/v2/staff/?search=jane+doe

ps: Somewhat related, if you want to create your own route, using data from multiple sites in the network: WP REST API – Multisite – get posts from all sites