Why Jace/Helaena isn't the foolproof solution it seems
My brothers in green, allow me to put forth my defense of Alicent’s rejection of this infamous marriage proposal, lauded as a possible solution to the Dance. Bear with me since this may get a little long.
Context
Rhaenyra proposes this to Alicent in episode 6 ("The Princess and the Queen") as a means of closing the chasm between their families.
Additionally, Rhaenyra offers that if Syrax brings forth a clutch of eggs, the dragon-less Aemond will have his pick of them. Viserys calls this a very handsome gift.
Alicent tries to extricate herself from the situation, so as to not refuse Rhaenyra outright. She tells her she will consider her offer and, luckily for her, is provided with the perfect exit opportunity - Rhaenyra's clothes have stained with breastmilk.
Now, marriage proposals are an issue to be considered carefully, not a decision you make on the spot or overnight, so Alicent is right to exercise her right to "think about it", however, I will grant that the scene immediately after shows that she has absolutely no intention of accepting, at least not in that very moment. Could she have changed her mind after sleeping on it? Let's see.
Rhaenyra has just given birth to her third child, who looks nothing like her husband Laenor and everything like her paramour Harwin Strong. Courtiers have started whispering more loudly than ever and Rhaenyra feels threatened by this.
What does Team Black gain from this?
Jace benefits from Helaena's legitimacy and quells at least some rumours about his bastard status. If The Queen herself is willing to betroth her only daughter to him, it must mean that he is legitimate, right?
A dragon's egg is no great loss for Rhaenyra, she has a female dragon actively laying eggs. There will be others if she has any need. This move costs her next-to-nothing.
I will give Rhaenyra the benefit of the doubt here and accept that she was genuinely trying to mend the rift between her and Alicent's side of the family. However, I will suggest that, up to this point, she has already dug herself in such a hole, that right now all she can do is damage control.
So let's see why Alicent might refuse her, apart from petty reasons.
Jace's legitimacy
The main issue here is that, unfortunately, Jace is so obviously not Laenor's son that everyone with eyes knows him to be a bastard. Rhaenyra has tried to transform the court in this theatre of the absurd, where no one is allowed to state the obvious, but this is not something that can carry on indefinitely. I am aware that people in Westeros don't know about genetics, but they are not that unaware. It's not just that Jace doesn't look anything like his white-haired parents, it's that he looks 100% like Harwin Strong's child, who is standing next to him and Rhaenyra. There are cases in which the family resemblances is so strong (heh) that little can be done in trying to deny it. No wonder Alicent feels like she's going mad.
With all of the good intentions of Rhaenyra, Laenor and Viserys, who play along in this ruse and pretend Jace is a Velaryon, the threat of a succession war starting over his right to rule is very real.
This goes beyond the politics of House Targaryen. People often argue that Jace is Rhaenyra's son, not like Cersei's children, who are not Robert's, the actual monarch.
Cersei reasonably gets away with this because her children at least look like her. Rhaenyra's children look nothing like Rhaenyra AND nothing like Laenor. They, instead, look very much like Harwin. It's complete clown behaviour.
The problem is that this threatens the entire system of inheritances in place in Westeros. Bastard children do not inherit for a reason. Marriage is a political and economic alliance between families, where both families are very much invested in the concept that their accumulated wealth / titles / lands / power is transferred to someone who is actually related to them. This is a world without proper contraception, where adultery carries the real threat of pregnancy. The two families involved essentially enter into an agreement that has very real economic consequences (dowries, favours, different types of exchanges) and no one wants to endanger this arrangement because of some random child fathered irresponsibly. One party will always be wronged in a situation like this.
Which is why the issue of Jace's legitimacy should be a big deal for the lords of the realm as well. If it suddenly becomes OK for bastards to be placed in the line of succession, then many of them might get in big trouble. Many conflicts of succession might arise because of this, maybe even wars.
Of course that, as a modern audience, we think it's super unfair that Jace doesn't get to inherit just because he was born out of wedlock. Society has progressed since then, thankfully, with the development of women's rights and family law and, of course, contraceptive methods allowing us, for the first time in history, to choose when/if we have children and how many of them. But Westeros doesn't have that context. So, in order for people to not kill each other endlessly over inheritances, they had to implement SOME rules and settled over male primogeniture. Completely fair? Of course not. But at least it offers some degree of stability.
The effects of this can be seen directly when it comes to the inheritance of the Driftwood Throne. The rest of the trueborn Velaryons (not just Vaemond) have genuine beef with the idea of Luke being ahead in the line of succession. No one else in Westeros gives up their inheritances like this, so why should they?
Therefore, why should the people of Westeros accept a ruler that doesn't respect the laws of the land or one who creates special exemptions for themselves, without putting in the proper work to reform society and make it a little more fair? This is a flagrant breach of the Westerosi version of the social contract.
Therefore, I humbly imply that there will be a lot of disgruntled people, upset that the monarch (Rhaenyra) is disregarding the law and fucking up their entire political-economic system of inheritance.
Helaena
Jace's rule will, therefore, most likely be contested. Wars may even be fought over it.
By marrying her daughter to a bastard claimant, Alicent could very well be placing Helaena and her future children in danger. Violent conflict tends to not be kind to women. Remember what happened to Elia Martell and her children? That came to pass because it was far more expedient if Robert Baratheon didn't have any other rival claimants to this throne. People like to pretend that this is an imagined threat, in order to demonise Alicent, but it is not.
That being said, Helaena may very well die due to illness or in childbirth. Unfortunately, medieval-style medicine was not kind to people, especially women. What happens then to this great alliance? Alicent doesn't have a replacement for Helaena to offer up to Jace. What if she dies childless? I would argue this is not that far-fetched of a possibility.
Helaena will essentially be a hostage for the Blacks. Rhaenyra doesn't have a daughter to give to Alicent, so that they could be even steven. From Alicent's POV, she is basically handing over her daughter to someone she doesn't even trust.
Alicent's sons
Alicent has three living sons. That's a lot of options for people to rally behind, if they have a problem with Rhaenyra becoming Queen.
Why does everyone believe that Aegon/Aemond/Daeron would be safe, just because Helaena would be married to Jace? Helaena couldn't do a damn thing to stop their execution, if Rhaenyra's claim was contested.
Even if you want to argue that Rhaenyra could never do such a thing, Daemon definitely would order hit jobs on them or murder them himself if necessary. He really is That Girl.
This is, of course, supposing that Rhaenyra does end up marrying Daemon in this scenario as well. I would argue it's a very real possibility, since they are so obsessed with each other.
Dragons
As it stands in Ep. 6, Team Black has 4 dragons at their disposal (Syrax, Seasmoke, Vermax, Arrax), with a new dragon egg for Joffrey that will later hatch into Tyraxes, making that 5. There is the possibility of Meleys (?), if push comes to shove. It's debatable whether Daemon & Laena would join her side, but it's worth mentioning they themselves have 3 additional dragons (Moondancer, Caraxes and freaking VHAGAR). If Rhaenyra keeps having children, they most definitely will have dragons.
What does Team Green have? SUNFYRE. Whom Aegon bonded with as a hatchling, making him young and small. Daeron is in Oldtown by now, so maybe he has already bonded with Tessarion, another hatchling. That brings Team Green's dragon count to a pitiful 2 baby dragons with inexperienced riders.
Helaena doesn't have a dragon in episode 6, because Aemond is seen going towards Dreamfyre. He would not be doing that if he knew Dreamfyre was bonded to his sister, it would be illogical. For someone as dragon-obsessed as he is, he definitely would know that.
So what happens between episodes 6 and 7? Helaena must have claimed Dreamfyre. At the end of the episode, we see 3 dragons flying back towards King's Landing. One of them is obviously Sunfyre, one is the newly-acquired Vhagar, Daeron is in Oldtown, so the other one must be Dreamfyre.
We know from Aegon and Aemond's conversation in Ep. 7 that Aegon and Helaena have been betrothed. Why? I would wager one of the reasons is that Helaena turned out to be a dragon-rider herself.
There is no way in hell Alicent would be giving Rhaenyra Helaena, alongside Dreamfyre, her biggest freaking dragon (remember Vhagar has not been claimed yet).
As it stands, when Alicent approved of the betrothal between Aegon and Helaena, her side of the family had two small dragons and Helaena's DREAMFYRE, quite a sizeable beast, a huge win for Team Green. Ain't no way Rhaenyra is getting her hands on her daughter now. Dreamfyre was Princess Rhaena's dragon, the Queen in the West/East. She should be larger than Syrax and Seasmoke, even Caraxes or Meleys, more on par with Vermithor and Silverwing.
You know what else Dreamfyre is? A female, egg-laying dragon, who has resided all this time in the Dragonpit. Which makes Rhaenyra's additional offer of a dragon egg moot. Why would Aemond need eggs from Syrax, when he already had access to eggs from the unclaimed Dreamfyre? Remember, Dreamfyre laid the egg Rhaenyra chose for ther brother Baelon - the same egg that Daemon stole, so it's not as if she's hit dragon menopause. Jace and Luke are shown to have picked the egg for Joffrey themselves from the Dragonpit; it is not specified whether this is Syrax's egg. If necessary, Viserys is still King, he can order additional eggs to be brought from Dragonstone for Aemond to pick.
This has already turned out very long, but, all in all, I wanted to point out the flaws in Rhaenyra's plan regarding Jace and Helaena's possible betrothal. I appreciate the fact that she was trying to remedy the rift, but she failed to understand why this deal was not exactly the home-run for Team Green that she intended. A very good arrangement for her, but not necessarily great for Alicent's peace of mind.