The ability to do so is important, because proof is central to mathematics. Those who can't distinguish between good argument and bad argument certainly ought not to be teaching mathematics. Even those who *can* do so but don't understand that proof is central to mathematics (and this includes many who mistakenly think that they want to major in mathematics) ought not to teach mathematics. (And the reason, I suspect, that the latter category includes so many who discover too late that they don't really want to study mathematics, is that too many of them make it through to become mathematics teachers. Allegedly, anyway.)
The aim of the test seems to be to test whether students can differentiate good arguments from bad arguments.
The aim? The target is fine (as you wrote above), the aim is very poor. A survey format?
I would have asked...
1. Is this proof correct?
2. If not, why?