Completeness of the strict binary relation
Your counterexample looks correct.

But an even simpler example would be $X=\{a\}$. We have $a \succsim a$, so $a \not \succ a$.



More generally, we usually define


$\succsim$ to be reflexive (i.e. $a \succsim a$ for all $a$); and
$\succ$ by $a \succ b$ if $a \succsim b$ and $b \not \succsim a$.


By the above definitions, $\succ$ cannot be complete on any non-empty set $X$ because for any $a\in X$, we have $a \not \succ a$.