In episode 3, many argue that Prince Aegon should be the next ruler of Westeros; others argue that Rhaenyras should remain the throne, as he is the actual firstborn son of King Viserys. He wants a clear sign of the right direction and sees a white stag running around the Kingswood.
When Viserys (Paddy Considine) is called to a stag, it is not what he expected. It is, as one of the helpers notes, still a big lad, but the animal is not white. This moment is staged, underwhelming, and lacking the clear symbolism he so clearly seeks, provides no clarity about who the gods wish to show their favor.
Rhaenyra (Milly Alcock) is the one who spots the huge white stag at a cliff in the morning. It is hidden by design, but the whole episode feels a bit mystical, tapping into the type of magic Game of Thrones used to employ to keep its high fantasy characters guessing.
The magic in the world of A Song of Ice and Fire is unpredictable and randomly spreading throughout the world. No religion had a clear grasp on magic, yet none of them had a clear grasp on it. There were changing faces, reanimation, and ghost demon babies, all of which existed with the same certainty as lightning or a vision from looking into a fire.
House of the Dragon is a book based on Fire & Blood. The book itself is a fairly dry recounting of the events as told by three individuals who witnessed (or witnessed) them. While George R.R. Martin brought whole characters back to life in the shows that were left dead, Fire & Blood reads more like a textbook, leaving out the more otherworldly elements of Martin's world.
The stag hunt in episode 3, like Aegon the Conquerors' dream from episode 1, is a step back towards the more mysterious world of Thrones. This time, the story may be one we know the end of, but the signs along the way are more ambiguously otherworldly. The Iron Throne seems to reject certain occupants, and the white stag appears to those who may be worthy. However, is any of that the best measure of good ruler?