If you recall, Rhaego was supposed to be "The Stallion that mounts the World". And he was never born.
Hence, there is the chance that the "3 heads of the dragon", "Azor Ahai Born Again" are red herrings. Maybe Jon could have been Azor Ahai born again... until he was killed by the Night's Watch.
Maybe there does need to be 3 heads of the dragon to conquer westeros, meaning that with only 2 targaryens it is not possible.
The concept that a prophesy will come to pass irregardless of the actions of the characters goes against the type of story Martin is writing. A prophesy about a character suggests the character is safe until the prophesy happens. Since no character is safe, no prophesy can be safe from just not happening.