How can I use wp_after_insert_post with $current_screen?

Based on your comment, you don’t actually care about the current screen. You care about the post type. They might be related but they’re not the same thing. If you want to target a specific post type in the wp_after_insert_post hook, you can use the post object passed to the hook callback:

    function( $post_id, $post ) {
        if ( 'post' === get_post_type( $post ) ) {
            // Do something.

Just replace 'post' with whichever post type you want to check.