What you are getting to is the basic question of whether Chomsky is right or not. All non-trivial synthetic language are equivalent in expressive power. So, either this has some bearing on natural language or it doesn't. But it is proven true.
The reason automatic translation is easy to trip up is that humans (except those translating Japanese or Chinese product instructions) understand what they are translating - they understand both the original and the translated result. No machine or mechanical process can be said to "understand" anything. Which goes back to the more basic question: Is there something going on in your head, called "intelligence" that cannot be implemented in a machine?
I don't think it has to be intelligence. It's more like motive. Doing a little thought experiment, consider the word "dog". Does it have the same meaning to a child playing with a golden retriever as it does for a child who has been mauled by a pit bull?
In my humble opinion, meaning is an attribute of consciousness and cannot be tokenized. Human language is motivated by the intention of influencing other people's behavior. The deep meaning and purpose of language is not syntactical. Because we live in a culture dominated by written language, and actors and pundits reading scripts, it is easy to lose track of pre-literate language. A lot of what goes on in conversation is "between the lines."