Nothing and Nowhere

In the show, lucifer is very much underestimated due to his goofy, laid-back, stressed attitude and grief over lilith. He's not respected by the sinners, nor does he seem interested in the pride ring at all, resulting in him getting labeled as incompetent and leaving all the work to lilith, hense why the sinners seem to respect her that much while paying no mind to lucifer.

she rulled them, she asserted her power and empowered them through her songs and gave them hope, when they think of a ruller they think of lilith, because she's the one who has been looking after the sinners, that's why she's so influential to them, their her people. But lucifer absolutely despises them.

In Hazbin’s sister series, which explores all of Hell beyond Pride, Lucifer is surprisingly the only one frequently mentioned. People swear by his name, his throne remains a fixture in the council, and his presence haunts the narrative! (Even in heaven, but that's a whole other can of worms), but there is absolutely no mention of lilith, who presumably had all the duties.

He's very much still remembered in the other rings of hell, which means he had to be active for him to have left some sort of impact, we never see any sinner exclaim themselves with lucifers name while it's a whole other deal in the other rings

Each of these people remember a different ruler.

sinners and pride revere Lilith and acknowledge her as their true queen.

Hellborns + all seven rings + nobility reference Lucifer; Neither group even acknowledges the other spouse’s authority.

We actually see this reinforced twice — in Episode 5, and even more clearly in Mastermind!

with the sins asking for lucifer to come back (with gifts! Easily the highlight of the episode because holy shit mammon giving out money, you know their desperate) You don’t usually ask a man who “does nothing” to return.

in mastermind, when Satan starts talking about lucifer, no matter how brief, the sins still turned to glance at his throne (and if anyone noticed there's only one throne present, not two as you'd usually expect 👀)

Satan is the one in charge of the court, he's the one who handles the legal affairs of Hell because lucifer is absent now, it's a temporary position he took on when lucifer started neglecting his duties after liliths disappearance

It was bureaucratic, governmental, and systemic.

We’re talking laws, courts, rulings, governance — the whole structure of Hell’s civil order and Satan holding the seat because Lucifer is absent implies the MOST important thing, There is a seat. A throne. A role. A job, and it belonged to lucifer, and he runned it for a very long time, long enough for it to be engraved in all the rings.

he is also the one who gets Charlie a meeting with heaven in absolutely no time, so it seems he also used to handle celestial diplomacy as suggested in s1 when he tells Charlie to go in his place, and yet again when he gets her in heaven for an audience

So no lucifer wasn't just sitting on his ass for millenias. He just gave up and slowly retreated during the last 7 years.

But also add the fact that lucifer CAN'T harm sinners

If Lucifer:

Cannot harm sinners

Cannot punish sinners

Cannot discipline sinners

Cannot enforce law on sinners

Doesn't even care for sinners

…then OBVIOUSLY, he can not rule sinners. That's his punishment after the fall.

You can’t rule a population you aren’t allowed to touch.

So, who takes the role?

Lilith.

That's why she thrives in hell. The sinners are her people, and she had to actively earn their respect as their queen- as a sinner herself while lucifer stepped back because- he cant do so, and frankly, even if he could rule sinners I don't think he'd even be interested in doing so, he hates them.

Which leads me to believe:

Lilith was in charge of the sinners while lucifer was in charge of the hellborns

Which is why he fails when he eventually has to deal with sinners, their not his problem, they were lilith's and now that lilith is gone and he has to step back in after 7 years to help Charlie, he has to confront them and of course nothing goes as planned, because he never had to deal with them in the first place, they were never his responsibility before but they're surely now!

He handled the situation with heaven with absolutely no problems in s1, but now, in s2, he actually has to deal with the people he has been neglecting!

But that also begs the question of could lilith even leave the pride ring? If no, then this strikes a beautiful balance.

If she CAN’T leave Pride, then everything about their marriage suddenly becomes a lot clearer, Lucifer rules everywhere but Pride where he is cursed to watch the fruit of his mistake constantly without having the power to do anything— basically the only place he cant rule while Lilith rules all of Pride and it's people and comands their respect but nowhere else— because she cant rule beyond pride— their two monarchs trapped in different ways.

But then that makes the topic of their separation even more interesting

Lucifer agreed to the extermination of SINNERS…but secured a pardon for hellborns. Do you see it?

If sinners are lilith's subjects and hellborns are lucifer's then lucifer securing the pardon for hellborns make whole lot more sense (other than just protecting charlie) Lucifer protected his own people while Lilith had to watch her's get slaughtered annually, and he agreed to that system while having the final say on the matter. (Albeit forced, the situation is very complex, and there's always two sides to a story, which is what makes it so interesting, in my opinion!)

Is it his fault?

Not entirely — Heaven backed him into a corner, and he saved those he could. (But let's not act like he felt all that bad for the sinners)

But is the asymmetry real?

YES. Painfully so.

This is also why I think watching both helluva and hazbin is important while it isn't nesscary, the world building in helluva is still integral to hazbin and it is very useful when we need to see the grand scheme of things.