How would you make them answer truthfully? Admitting to being tired either means you're showing personal weakness, or you're calling your superiors incompetent at scheduling. Neither possibility sounds good for your career. I think it would be better to measure reaction times, eg. by making them catch a falling ruler immediately after it's dropped and seeing how far it falls.
This is why good leaders create an environment where you can be honest about such things. You are basically talking about an inhumane system led by fear. Then again we are talking about the navy here...
This is part of the reason why they institute fatigue standards, to try to keep people from being over tired and still working, which led to people being hurt.
The hard part is that no one wants to admit that they are tired. Everyone wants to be "that guy" or "that girl" who is always ready to go. After all, most people Ive met in the military have a hard time putting themselves to bed because they have the next watch when its 0200 and we're pulling migrants off rafts or running helicopter ops to interdict $100 million worth of cocaine. During lots of operations you will have a lot of the crew stood up to run things, so balancing crew fatigue with surge operations is no trivial task. The SWO community could take a lot of pointers from the aviation folks - they are way better about making sure people get mandatory rest.