An example of of an information theoretically secure protocol that is not cryptographically secure
In order for information-theoretic security to imply computational security, you need to require that the simulator run in time that is polynomial in the running time of the real adversary. This is the standard definition, specifically to avoid protocols such as you presented in your question.

So, the answer is:


If you allow the simulator to be unbounded in running time (and irrespective of the running time of the real adversary) then information-theoretic security does not necessarily imply computational security.
If you restrict the simulator as I described, then information-theoretic security does imply computational security.


In my opinion, it makes no sense to allow the simulator to be unbounded if the real world party is (and this is the case in the real world). Therefore, I strongly argue that you must restrict the simulator as described.