I agree. First the movie is fine as it is and not in need of the main thing a remake would provide, fancy CGI. Secondly it derives part of its charme from being a fifties movie with gadgets like a ship that looks like a flying saucer or an old-school robot.
The theme of the movie is also pure fifties/sixties. The Cage also has features this lovely sci-fi-ish treatment of psychoanalytical material and that's why a remake wouldn't work either.
If such themes reemerge naturally in the 21st century in science fiction I won't object but forcing it is bad, not to mention that it might get ignored or ruined in a remake.