In order to search all post-types using the WP REST-API, which is not possible with the API, you can use this piece of code to search for all specified post-types. Replace all the 'POSTTYPE' parts with your CPT's slug, and it will be searchable in 1 query. You can also add additional CPT's. Using the ?embedded=true, you get all the embedded data!

WP REST-API search all (specified) post-types

<?php
add_action('rest_api_init', function () {

    register_rest_route('mvaessen/v1', '/search', array(
        'methods'  => 'GET',
        'callback' => 'search_all_endpoint',
    ));
});

function search_all_endpoint(WP_REST_Request $request)
{
    //first get all posts through the normal WP_Query, to make this determine the order of display
    $query = $request['query'];
    $posts_per_page = -1;
    $paged = 1;

    if (isset($request['posts_per_page'])) {
        $posts_per_page = $request['posts_per_page'];
    }

    if (isset($request['paged'])) {
        $paged = $request['paged'];
    }
    
    $query_args = [
        's'                => $query,
        'post_type'        => 'any',
        'suppress_filters' => true,
        'posts_per_page'   => $posts_per_page,
        'paged'            => $paged
    ];

    $query = new WP_Query($query_args);

    //prepare controllers for the required post-types
    $postController = new WP_REST_Posts_Controller('post');
    $pageController = new WP_REST_Posts_Controller('page');

    //To add additional post-types
    //$POSTTYPEController  = new WP_REST_Posts_Controller('POSTTYPE');

    $output = [];

    //loop through the WP_Query results, and get the objects as proper WP REST-API objects
    foreach ($query->posts as $item) {

        $request = [
            'id' => $item->ID
        ];

        switch ($item->post_type) {

            case 'post':
                $output[] = $postController->prepare_response_for_collection(
                    $postController->get_item($request)
                );
                break;

            case 'page':
                $output[] = $pageController->prepare_response_for_collection(
                    $pageController->get_item($request)
                );
                break;

            //To add additional post-types
            /*case 'POSTTYPE':
                $output[] = $POSTTYPEController->get_items($request);
                break;*/
        }
    }

    $response = rest_ensure_response( $output );
    $response->header( 'X-WP-Total', (int) $query->found_posts );
    $response->header( 'X-WP-TotalPages', (int) $query->max_num_pages );

    return $response;
}

//how to use: /wp-json/mvaessen/v1/search?_embed=true&posts_per_page=2&paged=1&query=looking+for+this