include user profiles in search results?

I haven’t an ready to use solution. However I think you should enhance the query, so that you have the field of the users inside this. I think the follow example demonstrate it more.

The two filter hooks are necessary and get an result for the query like this:

SELECT SQL_CALC_FOUND_ROWS wpbeta_posts.ID
FROM wpbeta_posts JOIN wpbeta_users
WHERE 1=1 
AND (((wpbeta_posts.post_title LIKE '%search_string%')
OR (wpbeta_posts.post_excerpt LIKE '%search_string%')
OR (wpbeta_posts.post_content LIKE '%search_string%'))) 
AND wpbeta_posts.post_type IN ('post', 'page', 'attachment')
AND (wpbeta_posts.post_status="publish"
OR wpbeta_posts.post_status="private")
OR (wpbeta_users.display_name LIKE '%search_string%') 
ORDER BY wpbeta_posts.post_title LIKE '%search_string%' DESC, wpbeta_posts.post_date DESC
LIMIT 0, 10

You can also read this query on your tests really easy via plugin, like Debug Objects or Query Monitor. The sql query is only an example and sure not the result to use it. You must play with them and include the hooks below to get the right result. The source get als only an example to add one field from the users table, the display_name. Maybe a sql “nerd” can help.

// Enhance the JOIN clause of the WP Query with table users.
add_filter( 'posts_join', function( $join, $wp_query ) {

    // No search string, exit.
    if ( empty( $wp_query->query_vars['s'] ) ) {
        return $join;
    }

    global $wpdb;
    $join .= ' JOIN ' . $wpdb->users;

    return $join;
}, 10, 2 );

// Enhance WHERE clause of the WP Query with user display name. 
add_filter( 'posts_where', function( $where, $wp_query ) {

    // No search, exit.
    if ( ! is_search() ) {
        return $where ;
    }

    global $wpdb;
    $where .= ' OR (' . $wpdb->users . '.display_name LIKE \'%' . esc_sql( $wp_query->query_vars['s'] ) . '%\')';

    return $where ;
}, 10, 2 );