Let's say I post a new answer to an old question, and that my answer is objectively better (imagine for the sake of the question) than any of the other answers. Is it considered bad etiquette to notify people by way of comments of the superiority of the new answer, or is it better just to let nature work its course?
If you notice a problem with some other answer, and you've posted an answer to fix it, then commenting on the other answer would be fine. A comment saying, "Your answer doesn't handle the case where the [...], but this answer [link] addresses that problem." is appropriate.
Just writing something like, "I've added an answer here, and it's better." isn't really an appropriate comment.
New answers to old questions are generally not better. What they usually are is someone promoting their blog, making a comment, or advertising. Answers that genuinely add something substantial to an old post are comparatively rare.
The usual responses apply. Downvote, flag as spam, etc.