Interesting question. I suppose some do, and some don't.

My DH's eldest adult daughter is a Golden Uterus. OSD has always had difficulty getting along with others, so as a kid she never did any extra curriculars, went to dances, or excelled at anything. She had one goal - to have kids and be a housewife. She loves her kids, but she is the sun and they the planets that orbit her. Her SSs all played football, so her son had to as well. Her stepsister's daughter grew up participating in cheer, so OSD's daughter had to, as well. Their accomplishments belong to HER, and she revels in it. She does take good care of them, but anger is her default emotion so she often shouted and berated them. We are estranged, so I don't know what the current status of her relationship with her kids is like or if she's still relying on them for narc supply.

Absolutely they do because narcs don't see Thier own faults. They project them on to others. So it's not them treating their children poorly it's you.

Also the Golden Child is viewed as the most perfect. Narcs revel in their accomplishments and blame others for their faults.

There are different types of narcissist. The type that Mothers usually fall under is the one who plays favorites with her children, putting one against the others and using the child as a means of reflection of themselves as long as they're doing well. My SS's bm is a classic mother narcissist and she ignores him unless it's time for a photo op, moty opportunity (meeting teachers/principals) or she needs a post for social media. Now the second these kids stop performing, the Narcissist parent will turn on them. 

The public perception must be one of high status of some sort. They insist on special treatment for their kids because they feel they deserve special treatment and probably feel like they didn't get enough of it.

So in public, in schools, extracurriculars, etc. they insist on special treatment for their kids. They hug them, they cuddle them, they proclaim how talented/beautiful/athletic/special their child is when they are really just projecting themselves through their child. In private is a whole other matter, they play favorites, place blame on everyone else and if that child doesn't perform in those special circumstances that mommy worked so hard to get them, you better believe they are told how ungrateful/untalented/ugly/unathletic and what a disappointment they are. If not that it was someone else's fault. Someone intentionally sabotaged all of mommy's hard work. 

For any sd/hubby meets even though i had a 1 & 2.5 yr old  at the time. Then despite all the abuse hcgubm antics, she conviced sd she had their best interests at heart and my husband needed to prioritise transferring some of his properties solely to skid when he hadn't even set up savings or a will to help protect our 2 little kids so their basic education and living expenses were taken care of when skids all benefitted from this.

i refused to participate in any skid meets until such time hubby brought them into line as they continually disrespected me and made me feel unwanted when hubby just wanted to enjoy an outing with us too and wanted everyone to just get along.

i didn't want to spend these meets with skids ranting non stop about hcgubm.

She may not treat SD as well in private as she does in public. BM here was MOTY in the public and basically ignored SS in private.

I think it depends, in my case my exh was an extreme narcissist and to him only HIM mattered, his kids never did and still don't.  It was always all about his needs, his wants, his time, his life-his kids were of zero value to him.  I'm not sure which is worse, the narc that despises their own kids or the narc that makes them the golden child tbh.