My understanding (as a speaker of neither language) is that Japanese usage has diverged a fair bit from Chinese roots.

The phonetic bits, for example, are based on how Japanese speakers of the time thought the equivalent Chinese sounded, with allowances for Japanese and Chinese having radically different phonics. And then you do that a few more times because the language both evolved in parallel, and you end up in a situation where each kanji can have more than one phonetic representation based on the original Chinese pronunciation. It can also have a phonetic representation based on the Japanese word that means the same thing, which will invariably be something radically different since Chinese and Japanese aren't terribly closely related languages.

So while they might be using the same writing system, the fact that it's a fairly elegant writing system for Chinese doesn't imply that it's elegant in Japanese. Not any more than the Roman alphabet being a very good writing system for Latin implies that English's writing system isn't a bit of a mess.

The original article makes no mention of kanji (Han zi). Instead it ends with: "Well okay, I'll come clean. I've actually been secretly teaching Japanese to you. 口 actually really is the Japanese character for "mouth" and 食 really is the symbol for eating."

Not exactly. Those are Chinese characters and Chinese meanings. The article does not teach anything exclusivly Japanese. For anyone who was not familiar, they would go around with the impression that these characters really originate from Japan. You can imagine how the Chinese people or people who study Chinese would feel being shown this cool new way to learn "Japanese".

In that 口 article yesterday, several commenters seemed angry that the author referred to kanji as "Japanese characters".

So apparently this is some kind of thing, that some people take seriously. New one on me.

Kan Ji or Letters of the Han Dynasty