1. Cersei has a distinct advantage over Rhaenyra, since her children look like her. She can very easily argue that they favour her, as their mother, and this is exactly what everyone believes for years, including Robert. Since Jaime is the male version of Cersei, Joffrey, Tommen and Myrcella can look like no else. Catelyn's kids look like Catelyn and no one bats an eye. Only Arya and (to Catelyn's irritation) Jon look like Ned. However, Ned doesn't ever doubt his children are not his.

Rhaenyra's kids look nothing like Rhaenyra and nothing like Laenor. They, instead, share distinct physical traits with her sworn shield, a man seen very often in her presence. People are not idiots. There is no plausible deniability here. You can bet your bottom dollar that if Cersei's kids were, say, Dornish-looking, people would be calling her out for her bullshit.

There is a way you can reasonably get away with passing over your bastards as someone else's, but that is 100% not Rhaenyra's way. This is why Cersei is chilling in the Red Keep, living her best bad bitch life, while Rhaenyra is running away to Dragonstone when the rumours are nipping at her heels. They are not the same. There are no paternity rumours to quell Cersei's girlboss vibes. She is sly enough that even Robert is convinced he inseminated her (gross).

2. I'm not going to get into the intricacies of Ned Stark's Scooby-Doo, Hercule Poirot mystery plot of unraveling Cersei's misdeeds. Ned has his own beef with the Lannisters and is convinced they are up to no good. He investigates them like the meddling kid he is and comes away with a suspicion. He knows nothing (heh) for certain until Cersei verbally confirms it for him. yOuR bRoThEr Or YoUr lOvEr. boo!

Had Ned not been on the Lannister trail from the very beginning, a fair assumption can be made that he never even would have suspected anything untoward. He never questions the children's paternity when they visit in Winterfell.

Again, this is distinctly different from Rhaenyra's situation. No one believes Cersei's children are bastards,* whereas no one believes Rhaenyra's children are trueborn. Pretending otherwise is very, very strange.

*at the beginning of AGOT, at least

3. Robert claimed Joffrey all his life and specifically named him his son and heir in his will, under dictation, to Ned. In turn, Ned deliberately changed Robert's words and wrote them down as "my rightful heir".

This is a parallel to show!Alicent, who misunderstands Viserys' dying words and him naming his son Aegon as heir. If Alicent didn't have the right to muddle the King's meaning, then neither did Ned. However, no one in their right minds is arguing that Ned is a traitor to the Crown. I wonder why is that?

I have already pointed out the circular logic in arguing that Robert only said that because he didn't know the children weren't his.

4. So what does this mean? Can anyone just accuse anyone they don't like of being a bastard and, thus, endanger that person's entire social status?

No, of course not. But, unfortunately for Cersei, Ned and Stannis aren't just some randos in a tavern. Ned is the Hand of the King. Stannis is Lord of Dragonstone and on the Small Council. These two men have a stalwart reputation and are renowned for their obsession with justice, duty and, in Ned's case, honour.

If Ned Stark stands in front of the Iron Throne and proclaims Joffrey a bastard, risks his daughters' lives and literally ends up losing his head as a result of this,

if Stannis Baratheon sends letters throughout the realm claiming Cersei's children are illegitimate,

the people of Westeros are going to pay attention.

These two very important men using their public platform to denounce Joffrey and starting wars over this? Say what you will about them, but they are not oathbreakers and they are not liars. No, they don't come with DNA tests, but for a lot of Westerosi, this is enough. They believe it.

Is this foolproof? No, of course not! But it convinces enough people that they are willing to band together to support rival claimants to the throne, thus igniting the War of the Five Kings. Speaking of political headaches, this is a huge one!

That being said, while Cersei is playing in the Champions League, Rhaenyra is fighting for her life in the relegation zone. She doesn't even need a Ned or a Stannis to cast doubt on her because no one believes her kids are not bastards.

Moreover, Vaemond obviously parallels Ned in this story. He tells the truth in open court and loses his head for it. In the show, Daemon and Viserys play the same role as Joffrey. In the texts, Rhaenyra and Daemon are stand-ins for Joffrey. This is not meant to be a triumphant moment of girlbossery. This is an abuse of power and an act of terror.

All in all, I'm sorry to say, but Cersei wins this hands down. She is savvy enough in her choice of sperm donor and can maintain plausible deniability without looking like a goddamn clown and the entire circus to boot. She holds the capital and has access to all the emblems of state after Robert dies. In contrast, Rhaenyra is floundering across the Blackwater Bay, yelling at the dragon gargoyles that her children are trueborn.

Why is this issue important in the story?

a). No one has a problem with Jace being King.

If people had a problem with Joffrey being King, enough to go to war over it, it would be narratively inconsistent for them to just accept an obvious bastard as King. It would contradict the internal logic of the fictional world we're talking about. That's quite some level of suspension of disbelief just because some fans like Jace. This isn't about him being amiable or a good kid.

b). They're still Rhaenyra's sons / it's a Targaryen internal matter and concerns no one else / the concept of Jace being King doesn't personally affect anyone else, so why does anyone care?

Because it's the freaking law! The name of the crime Rhaenyra commits is high treason! Punishable by exile or death!

No, the crime is not adultery, it's not having bastard children, it's specifically putting said bastard children in line to the throne. In that, Rhaenyra is as guilty as Cersei is.

It absolutely does affect others, since Rhaenyra actively steals the inheritance of House Velaryon for Luke. How is that not a crime? I would even go so far as to say that Laenor and Corlys are complicit in it and should be punished as well.

Contrary to bafflingly-popular erroneous beliefs, the monarch can't just do whatever they want. Even in absolutist monarchies, the sovereign serves the vital social role of upholding the law and the rights of their subjects. Rhaenyra breaks said law by committing theft, murder, high treason and destabilizing the entire system of inheritance.

c) Rhaenyra breaks the social contract

Jock Locke argues for the "right of revolution" in the Second Treatise of Government. He writes that when the government acts against the interests of its citizens, then said citizens gain the right to overthrow it and replace it with an authority that will protect their interests.

I am not trying to impose 'progressive' understandings of the political process anachronistically, in a medieval fantasy; my thesis-statement is that we have already seen this concept at play within the world of ASOIAF: the Faith Militant uprising against Aenys I and Maegor due to their practices of incest and polygamy and Robert's Rebellion, caused by Rhaegar kidnapping a noble lady and Aerys II carrying out executions without due process. The people of Westeros are not unfamiliar with opposing monarchs who don't abide by the law.

The question of Rhaenyra having bastards is framed in a lot of commentary through the lens of her right as a woman to have extra-marital sex and not be demonised for it and to find fulfilling love within the constraints imposed on her by her station. While debating the personal individual freedom of women in a patriarchal feudal society is not to be side-lined, her fundamental fault is that she is demanding rights and exemptions for herself, while the rest of the country have to abide by an entirely different set of rules.

The laws of inheritance, as unjust as they may appear to our modern eyes, are in place to prevent crises of succession, violent conflicts or even large-scale wars from starting every time someone's estates are passed on. Illegitimate children suddenly gaining access to inheritances threatens the political and economical calculations that predicate many Westerosi marriages.

Imagine paying a handsome dowry for your daughter, just so her husband's bastard birthed by some high-born mistress to make use of his maternal family's resources and cheat your legitimate grandchildren out of theirs.

Imagine being married to some lord and now his random bastards threaten the inheritance of your lawful children. Because, hey, the Queen acts like this is fine! This is Catelyn Stark's worst nightmare.

You think you can just sue your husband? What a silly notion. You think you can sue the bastard claimants after your husband is dead? Tough luck, your liege lord may rule in their favour by taking a leaf out of Queen Rhaenyra's book. You think you can appeal to Queen Rhaenyra? How are you going to travel all the way to King's Landing? Good luck with that, maybe you're built different and don't die during this dangerous and expensive journey.

Is this fair for the illegitimate children? Hell no, but Rhaenyra and Viserys are not planning on reforming family law in any meaningful way, because they know what a hassle it would be and how much opposition it would meet!

It reeks of rights for me, but not for thee and I, for the life of me, don't understand the stronghold she has on the liberated feminist brigade.

and finally

d). The Green Coup is not dependent on the legitimacy of Rhaenyra's children.