In the video Ravi Zacharias is answering a question from the audience:
How can you present the Gospel to a friend who has been raped at a young age?
His answer prepared the audience with a brief philosophical position that the OP transcribed and then Zacharias addressed the "existential" issue by referencing Eleanore Stump's contribution to a book Philosophers Who Believe:
Her entire article - essay - shows how she came to know Christ by facing the problem of suffering head on.
Zacharias and perhaps also Stump are primarily approaching the topic of evil and suffering from a "head on", pastoral perspective. There is a corresponding philosophical perspective that Zacharias briefly touches on when he remarks "if you assume a moral law, you must ultimately posit a moral law-giver".
Is it really the case that the assumption of a moral law (or obligation) requires the assumption of a moral law-giver? The OP appears to be primarily concerned about this assertion:
What I'm stuck on is how he jumped from positing a moral-law giver to showing that God exists and hence makes the problem of evil inconsistent. A moral law does not require a God (from what I've learnt regarding ethics), so why is it that Zacharias' defense proves the problem of evil to be self contradictory/self destructive? Am I missing something important?
The existence of a moral-law giver would be the existence of God and Zacharias' theology has identified that God.
What one might be missing is a distinction between ethics and moral obligation. G. E. M. Anscombe in her article "Modern Moral Philosophy" would agree with Zacharias that moral obligation (law) requires a divine law-giver. However, she also claims that one could have an ethics without obligation by imitating Aristotle's ethics in which she sees no obligation only a judgement whether something is done well or not. One of her key points as Wikipedia quotes it is:
Concepts of obligation, and duty — moral obligation and moral duty, that is to say — and of what is morally right and wrong, and of the moral sense of "ought," ought to be jettisoned if this is psychologically possible; because they are survivals, or derivatives from survivals, from an earlier conception of ethics which no longer generally survives, and are only harmful without it.
Why we are in a situation where moral philosophers feel the need to go beyond Aristotelian ethics to include theistic obligation she considers historical. Between the time of Aristotle and today there was a theistic cultural influence "which no longer generally survives" that made obligation part of ethics.
At the end of her paper she claims that with modern moral philosophy one can "procure the judicial condemnation of the innocent". This is what makes modern moral philosophy "harmful" without theistic obligation. Her perspective may help clarify what Zacharias said that the non-theistic approach to the question of evil "self-destructs in terms of an objective rule".
Anscombe, G. E. M. (1958). Modern moral philosophy. Philosophy, 33(124), 1-19. https://www.pitt.edu/~mthompso/readings/mmp.pdf
"Eleonore Stump" Wikipedia https://en.wikipedia.org/wiki/Eleonore_Stump
"Modern Moral Philosophy" Wikipedia https://en.wikipedia.org/wiki/Modern_Moral_Philosophy
"Ravi Zacharias - Addressing the Problem of Evil" nlcfvideoteam https://www.youtube.com/watch?v=P9b0PJDDof4
"Ravi Zacharias" Wikipedia https://en.wikipedia.org/wiki/Ravi_Zacharias