#joannalauren's answer really is the best for this (unfortunately repetitive) question

[nobodysuspectsthebutterfly]

While I’d like to answer you, I’m afraid you have a lot of problems with your original question, which is a major reason you’re getting unhelpful results.

First of all, it is significant that it’s not only Rome that’s GRRM’s inspiration for Valyria and its Doom. While yes, ancient Rome (known-world-spanning empire with slavery and advanced technology such as its famous roads) was a reference, others include Atlantis (legendary magical/technologically advanced land destroyed in a cataclysm) [1] [2]  and the various Egyptian dynasties (incest to keep the blood of the gods pure, to consolidate power).[3] [4]  Atlantis in particular, in the fantasies of the sword-and-sorcery series and classic superhero comics that GRRM loves, has those rare escapees from its downfall, heirs of its magic and supernatural technologies.

Secondly, regarding the Targayen/Valyrian phenotype, GRRM realized (unfortunately too late), that it would have been more interesting if he had made the Valyrians black. Though he also admits that choice would have had its own issues, we can understand that it is only that their appearance is different that’s what matters:

...I wanted the Targaryens, and by extension the Valyrians from whom they were descended, to be a race apart, with distinctive features that set them apart from the rest of Westeros, and helped explain their obsession with the purity of their blood. To do this, I made a conventional ‘high fantasy’ choice, and gave them silver-gold hair, purple and violet eyes, fine chiseled aristocratic features.

So I strongly suggest you step away from concepts of “Aryan ubermenchen”, it’s hugely warping your thesis. If the Valyrians are “superior” in any way, it is not because they are blond or “albino” (which they are not; Bloodraven was the only actual Targaryen with albinism, and please note that particular phrasing has been frequently used to denigrate a character while also denigrating a disability). Their appearance is “a conventional high fantasy choice” and they could have looked like anything else had GRRM decided such. (Purple hair and silver eyes perhaps, but he probably wished to avoid such artificial “anime” looks much like he’s averse to DnD “dark elves”. Or perhaps dark hair and grey eyes, as Tolkien used for his own Atlantis-inspired Numenoreans, but GRRM decided to give that particular look to the Starks instead.) Mind you, the fact that this “conventional high fantasy” appearance involves light-colored hair and eyes and “aristocratic features” is very probably due to unexamined white supremacy in the fantasy genre, but nevertheless it’s not a deliberate “master race” reference by GRRM. (However, if you are interested in actual blond ASOIAF characters in relation to fascist imagery, I highly recommend this essay series.)

Third: I do not recall anywhere in the series or within TWOIAF or Fire & Blood where any Valyrians or Targaryens declare a “divine right” to rule. “Divine right” or being chosen by the gods appears nowhere in any mention of the Valyrian imperial conquest of Essos. Aegon the Conqueror, when he sent his letter to the Seven Kingdoms declaring himself the sole king of Westeros, did not cite the approval of any gods, and was crowned by his sisters as “Aegon, First of His Name, King of All Westeros, and Shield of His People,” while bearing a heraldic banner that Westerosi lords understood to mean he was now one of them, now following the customs of Westeros. And though Aegon and his dynasty sought the blessing of the High Septon of the Faith, so did the succeeding Baratheon kings.

Furthermore, if there is any sense of godly approval for Targaryen rule, it just puts them equal to other monarchs of Westeros and Essos. Notably, the Durrandon and Gardener kings claimed to be descended from gods, and other monarchs claimed descent from legendary heroes; and the ironborn believe their entire race to be “chosen of the Drowned God”. Also (since I’m genuinely confused by this assertion of Targaryen “divine right”), if you refer to the address “your Grace”, or the phrase “by grace of the gods King of Westeros”, again please note that is the title and styling of every monarch (and even lords), not specifically Targaryens. (And interestingly, while Daenerys does use the address “your grace”, the styling “by the grace of the gods” has never once appeared in her titles.) Or perhaps you’re simply misunderstanding the phrase “rightful king” or “by rights”? That does not mean “divine right”, but only “by justified claim”.

The only thing I can think that remotely approximates what you might consider divine right for the Targaryens is the Doctrine of Exceptionalism? But that declaration of the Faith does not state that Targaryens are superior, only that they are different, not the usual span of humanity and thus are an exception to the laws of the Seven that judge incest an abomination. Yes, it cites their atypical appearance and ability to ride dragons, and their origins in Valyria, but only in regards to the Faith tacitly ignoring their marriage practices, nothing to do with them being more suited to rule or chosen by the gods.

Also, it’s significant that while Fire & Blood put a name to the reason why the Targaryens could get away with incest while Jaime and Cersei’s children are called abominations, that book also showed the Doctrine to be flawed in many ways. First, the belief that Targaryen exceptionalism includes resistance to illness was tragically disproven with the death of little Princess Daenerys (and less called out, the deaths of Maegelle and Baelon, etc). Second, the Targaryen otherworldly appearance is unconnected to their ability to ride dragons, as seen with black-haired Rhaenys Targaryen (the Queen Who Never Was) and Rhaenyra’s brown-haired brown-eyed pug-nosed eldest sons, all three dragonriders. The dragonseeds also confuse the matter, since while many had the Targaryen appearance, their actual ancestry is not fully known and certainly was mostly peasant stock. Even Addam of Hull causes an issue, since if his father was Corlys Velaryon it’s notable that the Velaryons, while Valyrian, were never dragonlords (though apparently there was some Targaryen daughters who married into the family, but the last one generations before Aegon I). And it’s strongly implied that the dragonseed Nettles (“black-haired, brown-eyed, brown-skinned”) might have no Valyrian ancestry at all and tamed her dragon by acclimatization.

So. Issues with your question now dealt with, on to the question itself. What do the Valyrians represent, and what does Daenerys's conquest represent in re their legacy? To put it as simply as possible: the Valyrians were a magical people with notable gifts and powers unmatched by any in the world. They represent a lost age of wonder and terror, dreams and nightmares, beauty and horror, fire and blood. The magic they used to bond with their dragons (which may have involved atrocities), and the inbreeding they believed was required to keep it strong, left a legacy which was felt in their Targaryen descendants down to Daenerys herself. Their other legacy -- the slavery learned from Old Ghis, the hundreds of thousands enslaved to feed the Valyrian lust for power and precious resources -- was eventually the cause of their doom, with the birth of the Faceless Men within the mines. But Valyria’s greatest sin still lives on, and notably Daenerys has made it her mission to confront and end this legacy of her ancestors in Slaver’s Bay, and may well soon among the Dothraki and in Valyria’s children, the so-called “Free” Cities; and as well with Valyria’s cruelest fanboy, Euron Greyjoy. These legacies of Valyria given to Dany -- her deep opposition to slavery, and the magic within her -- as well as her rights and responsibilities to Aegon the Conqueror’s united Westeros, will all unite when she faces her destiny to confront the ultimate enslavers of humanity, the true enemy, the Others.[1] [2] [3] [4] [5]

If GRRM will give us any more meaning than that, well... we’ll just have to see what happens in the books. And let me just say, if you don’t wish to check all the links above but still want to see the best answer to a question (unfortunately) just like yours, I strongly suggest you read this excellent post. Hope this helps!

One question I’m still curious about in ASoIaF is “what are the Valyrians meant to represent”?

Because at face value they seem to almost be living affirmations of the concepts behind feudalism and divine rule. They’re this ethnicity of albino ubermeschen who tame dragons and who built an empire on sorcery and military power to the point where they seemed to literally warp reality to their purposes.

They were destroyed, but an offshoot of their people, keeping up the same social practices of divine right and massive incest - just minus the direct slavery - goes on to conquer an entire continent.

That, and the magic of the world seems to grow weaker without their dominance. Or at least without the presence of their mystical dragons, which only they can hatch and train.

I’m just looking for where this thread is going. Of course the Valyrians were not always GOOD rulers, in Essos or Westeros, but they are indisputably powerful and “important”, and Daenerys doesn’t look like she’s going to refute this destiny in the books.

I don’t think GRRM would be so stupid or weird to just have unironic ubermeschen in this story, but I’m wondering if - besides the Doom, the slavery, and a few bad kings - he really has made any refutation that the Valyrians ARE better than everyone else.