Oh Grey Warden, what have you done?

When Wei Wuxian wants to hide the fact that he's lost his golden core, he does it by putting on a show of arrogance, and this can only work as long as it does because ego is already such a big part of his personality. Young Wwx was already known as a willful, trouble-making rule breaker, so nobody's going to question it when he starts showing up to events without his sword. They might ask "what the hell is that kid doing?" but they can always answer their own question with, "Well he's Wei Wuxian. He's always been a disrespectful and done as he pleased."

Wwx never pretends to be anyone or anything but himself in his first life, but he dials up certain facets of "himself" to make the public think what he wants them to. Pretending to be the person that the outside world expects him to be makes a very good disguise, because it's against others' nature to question it.

And we can argue about how effective it is, but Wei Wuxian tries to do a version of the exact same thing when he gets brought back as Mo Xuanyu. He hears that Mxy was gay and a "lunatic" and says "well if you want insane, then you'll get insane." He leans as hard as he can into that public expectation, because if Mo Xuanyu is behaving like exactly the annoying, openly queer freak that everyone expects him to be, no one's going to wonder who else he might be.

Meanwhile, Huaisang uses more or less the exact same defense mechanism when he starts racking up things to hide. Based on his repeating school as a teen and late formation of his golden core, he presumably has a reputation from a young age as not the sharpest tool in the shed. People know him as the Nie brother who cares little for cultivation and developed far too slow to make use of his saber. To be unkind about it, he's a useless little dandy unfit to ever inherit his clan.

So when Huaisang wants to be sure that no one will suspect he's making moves behind the scenes, he leans into that and leans into it hard. He makes everyone think they're right—he is an idiot unfit to run his clan. But nobody's going to look twice at a fool, and nobody will suspect subterfuge of the head shaker.

Once again, though, Huaisang's act only works because people expect him to turn into a leader like the head shaker. The same act wouldn't have worked so well for someone like Wei Wuxian, because even though they disliked him, people knew he was talented and dangerous. Only Nie Huaisang can get away with playing useless for a decade, because he's playing as hard as he can into the worst of his established public persona. Others mistaking him for a fool lets him trick them into thinking that he is one.