#He's so committed to this that he was 100% okay with Daemon beheading Vaemond for questioning it

[stuffedeggplants]

Another aspect to this is that the expectations for how the ideal Westerosi noblewoman should act would heavily punish Rhaenyra even implicitly acknowledging that her Strong children are bastards, which the request to Viserys to legitimize them would do. Honor in Westeros is a pivotal cultural force that both men and women are beholden to, but for women, their honor is tied to their sexuality in a way that it isn't for men.

When it comes out that Daemon and Rhaenyra were "coupling" in a brothel, the main issue for our characters isn't incest or exploitation, it's that Rhaenyra may have lost her virginity (her 'maidenhead') out of wedlock and her honor and value as a woman along with it. Marriage is a political tool; if all of Westeros believes that Rhaenyra has slept with a man outside of marriage, it could be more difficult to marry her off into the most powerful, prestigious houses. This isn't an abstract thing-- by losing the opportunity to form alliances with these houses, you lose access to the resources they command. (Specialized military units, strategic geographic features, the economic power of major trading hubs and shipping lanes, rich farmland that can be worked year-round, etc.) Like Viserys said, perception is everything, and regardless of what actually happened between Daemon and Rhaenyra in the brothel, if the elite class believes that Rhaenyra has acted inappropriately and lost her honor as a woman, the political consequences are real and her reputation and value have been diminished forever. This is not a good look for Rhaenyra as the heir to the iron throne, especially considering how much she's fighting against the gale of cultural expectations already by being a female heir. Her honor needs to be unimpeachable, or at least appear to be so. People can question it privately, but speaking it publicly is a completely different matter because it destroys the image that the ruler is trying to project.

If Rhaenyra asks Viserys to legitimize her children by Ser Strong, it's essentially a public confession that they are indeed bastards. It tells all the major power players that she has broken with the restrictive norms of female sexuality, and has broken the oath of fidelity to her husband that is the marriage contract. Keeping one's oaths is another source of honor and personal respectability, and because Rhaenyra was born female, admitting her sons are bastards is doubly worse for her because of how tightly honor, respectability, and the legitimacy of her as a monarch are bound up with expectations for female behavior/sexuality. Pretty much everyone knows that it's nonsense and that her children actually are Ser Strong's, but with how their social dynamics work, there's no benefit to openly speaking the truth unless you want to delegitimize Rhaenyra and House Targaryen for your own benefit. If you want Rhaenyra on the throne, you need to be furthering and maintaining the perception, if illusory, that Rhaenyra is an honorable woman by Westerosi standards. Her sexual behavior and fidelity to her husband cannot be under question.

Tl;dr - The Westerosi concept of honor is tied to patriarchy and double-standards for female sexuality in such a way that it makes it impossible for Rhaenyra to even implicitly acknowledge the truth of her sons' parentage if she wants to maintain legitimacy as a female political actor and (future) monarch.

I've seen a number of posts both on this website and others saying something to the degree of "Rhaenyra could have just asked Viserys to legitimize the Strong boys!" And while I'm all for being critical of Rhaenyra's decisions, this is not the cure-all that it's made out to be. So what am I gonna do? Write a wall of text about it!

Here in the HotD fandom, people are much more inclined to defend Jace, Luke, and Joffrey Velaryon than they would Joffrey Baratheon or Ramsay Snow. But because prejudice towards illegitimate children is so foreign to most Western viewers, I've noticed that the default seems to be parroting the talking points of their preferred side: either their Targaryen blood is all that matters because Laenor is fine with them, or they're proof that Rhaenyra is unfit to rule because she's trying to pass off Harwin's sons as legitimate.

The "just legitimize them" folks seem to come from both groups, painting it as an easy solution for her problems or calling her stupid for not doing so. They recognize that illegitimacy is a much bigger problem in Westeros than it is for us, but this argument pretty significantly underestimates the scope of that problem.

In the books, Rhaenyra had an extra measure of plausible deniability. Rhaenys was dark-haired, and her sons could theoretically have inherited it from their grandmother. In the show, their race makes it very obvious that they are not Laenor's. Instead of arguing something that is clearly a lie, why not go to Viserys and ask him to legitimize her sons? If Rhaenyra and Laenor presented a unified front on the matter, what's the issue?

Well, asking Viserys to legitimize Jace, Luke, and Joffrey would mean admitting that they're bastards in the first place.

Just as it is treason to question the legitimacy of trueborn heirs, it is similarly treason to pass off bastards as legitimate heirs to the Iron Throne (see: Cersei and her children). Viserys has his head pretty firmly buried in the sand on this issue - his horse anecdote to Alicent is pretty good evidence that he's convinced himself it's a quirk of genetics, not that Rhaenyra is lying, though one could argue that he does know and is just choosing not to say anything (see: how he publicly accepted Rhaenyra's story regarding Daemon and the brothel, but sent her moon tea afterwards). If she went to Viserys and said she's been lying for years and would like her sons to be legitimized please, he would probably pardon her for treason and do it without question, but this is the kind of crime that people get executed for. Massive risk if she's wrong.

If the matter could remain between Rhaenyra, Laenor, and Viserys, then the risk is much lower, but for legitimizing her sons to mean anything, it needs to be public. That means admitting publicly to the lords of the Seven Kingdoms that she has spent years knowingly committing treason, cuckolding her husband, and planning to install illegitimate children as heirs to the Iron Throne and to Driftmark. This is an extremely bad look for the heir to the Seven Kingdoms.

The safest moment to ask, with the least social backlash, would have been when Jace was a baby or small child, when she could call it one horrible mistake and play the penitent who went to her father as soon as she realized he wasn't Laenor's. This wouldn't work more than once, so she would have to take precautions to ensure she has no more children by Harwin. By the time Joffrey is born, that ship has long since sailed.

So let's say that at some undetermined point after Joffrey is born, she goes to Viserys and gets her kids legitimized in a big public ceremony, maybe including a vow to uphold Jace's claim like the one the lords once swore to her. They're legitimate now, and there's a pardon for the associated treason. Why wouldn't this solve her problems?

Many of Rhaenyra's allies are those who see her as the lawful heir, and see supporting her as the right, honorable thing to do. If she admits to years of infidelity and treason, she's going to alienate large portions of that group for moral reasons, pardon or no pardon. It's a serious propaganda loss that gives the Greens plenty of ammunition to accuse her of being a promiscuous lawbreaker. "If she's admitting to treason, what else is she doing that she won't admit to?"

Then there's the fact that conveying legitimacy onto Rhaenyra's sons is not necessarily enough to clear the stigma attached to their birth. A Wiki of Ice and Fire does an excellent job of summarizing it:

There is a certain stigma that comes from being born as a bastard. They are said to be born from lust, lies, and weakness, and as such, they are said to be wanton and treacherous by nature.

They're treacherous "by nature." Is that a ridiculous belief? Yes, absolutely. That doesn't mean it's not a belief that holds sway in Westeros, that will continue to affect her sons. Even legitimized bastards are not fully accepted (see: the assassination attempt on Alyn Velaryon, the bad reputations for all of Aegon IV's Great Bastards, and the admittedly deserved hatred for Ramsay Snow even after he became Ramsay Bolton.) Daemon Blackfyre is the sort of exception that proves the rule - he gathered support through immense personal charisma, but his supporters also weaponized accusations of illegitimacy against his rival Daeron II. While Rhaenyra's sons already face plenty of prejudice as suspected bastards, she's not making their lives any easier by confirming it. Your average Lord Joe in (for example) the Westerlands, who's never met a Velaryon or a Strong, could have shrugged and dismissed the accusations; that possibility is now destroyed. A woman on the Iron Throne is already a hard sell, and a legitimized bastard on the Iron Throne is an even harder sell.

Finally, it would cost her the support of House Velaryon, one of her greatest allies, and remove the Velaryon name from her sons. The Strong boys would be legitimized as Jace, Luke, and Joffrey Targaryen, because of their mother, but once you acknowledge that they have no blood relation to Laenor, they don't get to be Velaryons anymore (no, their great-great-great grandmother Alyssa Velaryon is not enough). That's the way legitimizing bastards works in Westeros, they are treated as though they're trueborn children of their parents (in this case, Rhaenyra and Harwin). There's no real precedent for adoption the way Rhaenyra and Laenor would want to do it.

Corlys was willing to let Luke inherit based on the principle that "history remembers names, not blood," but once you establish for the history books that they're very much not Laenor's and Rhaenyra was cuckolding him (the fact that it's consensual would make it even more embarrassing for House Velaryon, because they live in a culture of toxic masculinity and homophobia), then Corlys is going to drop his support.

Rhaenyra could try to force the Driftmark issue, but she would be playing an extremely dangerous game. In Westeros, you can't just inherit things willy-nilly, there needs to be a connection to the seat in question to begin with (see: Ramsay Bolton needed to marry "Arya Stark" in order to get Winterfell, Robert Baratheon's ascension justified through his Targaryen grandmother). Naming heirs works for when claims start getting messy (see: Daemon vs Rhaenyra, Baela vs Luke-as-Laenor's-son, Robb Stark's heir) but there needs to be a reasonable relation to begin with, and "Because Laenor Said So" is unlikely to pass muster among House Velaryon or most lords of Westeros. If Viserys orders that Luke shall inherit Driftmark without any blood relation, he's setting a dangerous precedent that's going to make just about every lord nervous about losing their own holdings. When you're relying on the lords to support your claim, it's not a great idea to alienate them by reducing their power.

So there you have it. Rhaenyra could theoretically have Viserys legitimize her sons, but it would come at the cost of support for herself and her sons' claim to the Iron Throne. Insisting that her sons are trueborn Velaryons doesn't look like a great strategy, but I think that for Rhaenyra personally it's the best of two bad options.