As a matter of curiosity, what precise definition of “the Universe” is this argument employing?
Space and time started with the Big Bang. But did “existence”? How would we even know? Do we even have the conceptual tools to discuss it? Does it make sense to ask if something existed before time?
While I think that “begins to exist” needs to be cleaned up I don’t think your changes actually improve the argument. It is better to just note that the premises are in question.
I always thought Kalam was beside the point; If God wanted there to be unambiguous proof of His existence, then it would be a very simple matter to prove such proof. In fact, I think it would be difficult to avoid leaving incontrovertible proof laying about the Creation simply by accident. I suggest a corollary (?) to Kalam; God does not wish to be conclusively proven to exist.
Not directly on point, but related, Michael Shermer just interviewed Kieran Fox, a neuroscientist and author of I Am a Part of Infinity: The Spiritual Journey of Albert Einstein. Looks like a worthwhile read. A bit more weighty than Craig’s 11th century Persian syllogism…