Cannot get ‘sanitize_callback’ to work for rest parameters

You’re not getting the TESTING because your playlist argument should actually be in the args array like so: (reindented for brevity)

register_rest_route( SoundSystem::$rest_namespace, '/playlist/new', array(
    'methods'             => WP_REST_Server::CREATABLE,
    'callback'            => array( __class__, 'rest_add_playlist' ),
    'permission_callback' => function () {
        return is_user_logged_in();
    },
    'args'                => array(
        'playlist' => array(
            'description'       => __( 'JSPF playlist data', 'soundsystem' ),
            'type'              => 'string',
            'required'          => true,
            'sanitize_callback' => function($value, $request, $param) {
                return 'TESTING';
            },
        ),
    ), // end args
));