Call me old, or whatever, but personally, I think smilies and emoji in WordPress should be enabled by plugins or themes, they should not form part of the WP core system. There, I’ve said it.
Anyhow, when attempting to disable the emoji in my theme, I came across an issue whereby the option to disable the conversion of :) to images was missing in the WP admin area. The problem was, I didn’t know it was missing, so when I tried to implement this, I was still seeing emoji in my posts and comments. Annoying.