I wonder why narcissism leads to favoritism. I've seen a few models of what the behavior could be accomplishing, but what's the link between it and narcissism? Why does the narcissist have the identification/rejection pattern with their dependents? Why is it necessary to them for a rejected individual to exist? That may be the wrong question.
I would think, "It's so that they have someone to project their flaws against," but they do that with almost everyone. The narcissist doesn't need a special target to project their flaws. I think the SG is less unusual than the GC. The golden child gets treated uniquely by the narcissist. They're not that nice to anyone else.
Perhaps it's the narrative. The narcissist always believes themselves to be an especially good, smart person in a world of EVIL STUPID. Whatever they do is what an especially good, smart person would do. Most of them aren't insane enough to say they're wiser than any man or god ever invented, but they definitely believe in their own wisdom. So... When they have kids, that was obviously an especially good, smart act, too. Except that kids are messy and difficult. Sane, well-adjusted people accept that this is simply how reality works. Narcissists see it as evidence that kids come straight from the EVIL STUPID. Except that abusing their children creates a huge amount of cognitive dissonance. It puts them into a position where they can almost see the gaps in their narrative. The world surrounds them in better parenting examples. Other people, out in that world of the educated stupid, somehow manage not to beat their children even when their children do frustrating things.
The GC is the answer to this dissonance. The GC is proof that they're a good parent. The SG is just another EVIL STUPID who is conspiring against the narcissist; the GC on the other hand proves that the narcissist was right all along about everything.
I wonder if narcissists ever have one child who they spoil rotten. We probably wouldn't see those individuals on this forum. I also wonder if narcissists think to themselves, "I'm so glad I'm a better parent than my own parents were. My own parents could never understand how hard I was trying to please them, but I'm not making that mistake. Look at how happy, well-adjusted, and parentally respectful my GC is!"