Maybe moving to a specific instance of Rice's theorem is clearer. My point is that while he can't solve the halting problem—i.e. he can't reliably determine whether any arbitrary program will halt—an infallible programmer can reliably write his own programs such that they always halt.

So, to go back to the original point, you can't reliably determine the correctness of every possible program. But you could, if you were not prone to error like all humans, always write your own programs such that they are always correct.