In episode 3 of House of the Dragon, there isnt a lot of agreement on who should take the reign of Westeros next. Many support Aegon's second birthday (he is the firstborn son of King Viserys) while others insist the throne is still Rhaenyras (he is the actual firstborn)
The stag isn't what he expected, although it is, as one of the helpers cautions, still a big lad, but the animal isn't white. This moment, which is staged, underwhelming, and without the clear symbolism he so clearly seeks, gives no clarity about who the gods wish to show their favor to.
Rhaenyra (Milly Alcock) is the one who spots the huge white stag on a cliff in the morning. By design, the scene feels a bit mystical, tapping into the type of magic Game of Thrones used to enact in the future.
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 neither one of them had a clear grasp on it. All existed with the same predictability as lightning or a vision from looking into a fire.
House of the Dragon is based on Fire & Blood. The book itself is a fairly brief account of the events as told by three persons who witnessed (or witnessed) them. While George R.R. Martin brought whole characters back to life in the shows that left dead, Fire & Blood reads more like a textbook, thus misses the more otherworldly elements of Martin's world.
The stag hunt in episode 3, just like Aegon the Conquerors' dream from episode 1, is a step back toward a more mystical 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. But is any of that the best indicator of good ruler?