Can languages that are not closed against the Kleene star be regular?
No, your argument is not correct.

The "hence it's not regular" doesn't follow.  Not every regular language is closed under Kleene star.  Some regular languages aren't closed under Kleene star.  I think you've misunderstood the closure property: the closure property states "If $L$ is regular, then $L^*$ is regular".  Note that you can still have a regular language where $L \ne L^*$.

To learn how to solve this question correctly, you may want to check out our reference questions.  See especially How to prove a language is regular?, Show whether the language with almost as many 0 as 1 in every prefix is regular, Language where every prefix has almost equal a&#39;s and b&#39;s.