The existence of law implies a Lawgiver.
Not necessarily. You’re just filling in a gap in understanding without evidence.
Purely random events also do not create matter.
Why not?
Therefore it logically follows that the universe itself must have a cause
No it doesn’t. You are applying biological reproduction theories to the birth of a universe. That’s comparing apples with universes.
You are misrepresenting the "god of the gaps" argument. According to RationalWiki: "God of the gaps (or a divine fallacy) is a logical fallacy that occurs when believers invoke Goddidit (or a variant) in order to account for some natural phenomena that science cannot (at the time of the argument) explain."
The argument here is not for a particular natural phenomena, but for why there is something rather than nothing and why that something operates under a series of very precise laws. Science can explain the "what" of natural phenomena, but cannot explain the "why", or that which pre-existed the entirety of the natural universe because the scientific method cannot apply to that which cannot be observed.
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