When telling Rhaenyra that he's going to name her heir, King Viserys passes on a family secret, passed down from king to heir. Aegon the Conqueror has a dream about the Others coming, and so he conquered the continent in order to unify it to face the threat.

This is bullshit. First of all, it pretty much outright states in the books that Rhaegar figured it out on his own. But let's say it was just the Mad King, who had issues with his heir, declined to tell him. This does not add anything except make the whole Targaryen conquest an exercise in futility, since it is pretty apparent (in the books, at least) that Daenerys is going to play a role, even if she is highly unlikely to ever sit the Iron Throne, in fulfilling the prophecy, along with Jon, who similarly had nothing to do with his Targaryen bio-parentage. In other words, the Targaryens did not need the Iron Throne to fulfil Aegon's prophecy, nor the unity of Westeros. In fact, the Iron Throne arguably has crippled Westeros' ability to unite or effectively fight off the Others on the physical level, because of the epic, continent-wide wars that have devastated the continent with massive loss of life and property, all because there were now vastly greater stakes in winning a civil war. Before the Targaryens, the Gardners or Lannistes could have internal wars for their own thrones, and it would not affect the rest of Westeros. By erasing the borders, Aegon & family erased the buffers that protect each region from the political follies of the others.

Even worse, is that the extant show to which this story is a prequel, shows that the Targaryens were not even necessary to the defeat of this enemy. It was a Stark who struck the killing blow, on her home turf of Winterfell, in a scenario set up by a Stark shaman, with a distraction from a Stark-raised fosterling. And given the price the Starks pay for having a Targaryen overlord, repeated throughout history, arguably, the greatest effect the Targaryens had was to undermine and hamstring the REAL line of saviors.

But lets put all that aside and look at Aegon's Conquest: He did nothing to bolster the realm's defenses against the threat he was acting to prevent. He just left the Night's Watch in place and the Starks in the North carrying on as they had for millennia. He did name them hereditary Wardens of the North, but he named other Houses Wardens of the other cardinal directions as well, who might consider their positions exempt them from obeying the Starks as they attempt to rally the realm against a threat from beyond the Wall. The Red Book of Westeros even makes a point of mentioning that Aegon more the most part, left the constituent parts of the realm alone, leaving unifying the legal code for his successors (and none of the men he might have prepped for the job, from his brother Maegor, his son Aenys to his elder grandsons made any efforts in that direction). He created no institutions to unite the realm, aside from the monarchy, where he not only left the old kingdoms more or less intact, but strengthened their individual rulers in some ways, by making them Wardens.

Jaehaerys must have passed on this secret to Viserys, but despite having Septon Barth, an authority on the paranormal as his Hand, did very little to work toward the threat. Yes, his Queen strengthened the Watch, but at the expense of House Stark, the next line of defense. That's not bolstering defenses, that's just rearranging the current ones. And at the beginning of his reign, he weakened the Watch by sending all the religious dissidents who rebelled against his uncle (which Jaehaerys ALSO did, but they were bad and he was good) to the Wall, where they overthrew the Lord Commander of the Watch, and suppressing this rebellion cost the life of the Lord of Winterfell. The Targaryens made no efforts to bolster the economy of the North, by investing in its development, or stationing royal garrisons to augment the Night's Watch or raising funds for the Watch from among the lords of the realm. Rather than move their court to a more central location, or one more accessible to the North, they hunkered down near their family seat of Dragonstone, and all the efforts of building up the monarchy suggest establishing a Targaryen cult, rather than a multi-generational effort aimed at a specific threat.

Likewise Viserys, the only one we can absolutely say believes in, and takes seriously this prophecy and House Targaryen's duty to work for it, did nothing in the books to work toward this goal, and does nothing on the show, giving every bit of his on-screen attention to his family, whether funding his brother's police project, to throwing a tourney to celebrate his forthcoming male heir, to judging his brother for lese majeste to naming his daughter as his heir. Even when other topics are raised in the Small Council, he is primarily concerned with members of his family.

Basically, this is another TV show cut-and-paste insertion, where they stuck in something because "we thought it would be cool" without giving it any other thought.