Could this be something similar to the principle that any language capable of doing a short list of things is Turing-Complete and can represent anything any other Turing-Complete language can do? That is, might it be that preventing our language from having more potential than we can use requires extra, arbitrary restrictions?
EDIT: I meant to also say, "and to work for the purposes we need it for, human language has to be able to do all those things."
You don't necessarily add restrictions to a language to stop it from being Turing-Complete, you can just not give it the necessary axioms or whatever. I mean, in a regular language, there's no rule saying 'you can use all these regexps and atoms unless you're using them like this, because that would be Turing-complete'.
For a human example, look at the reports about the Piraha language. It's not that they ban recursion out of superstitious dread of the infinite or something - it's apparently that they simply don't understand it/use it.
[I'd put this in an open thread, but those don’t seem to happen these days, and while this is a quote it isn't a Rationality Quote.]
— Geoffrey K. Pullum, Language Log, “Never fails: semantic over-achievers”, December 1, 2011
This seems like it might lead to something interesting to say about the design of minds and the usefulness of generalization/abstraction, or perhaps just a good sound bite.