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.