Simba gets special treatment not because he's the king's son, but because he's the only cub out of his litter who survived.

Lions are typically born in litters of two to three cubs, a fact that's confirmed in canon by the fact that Nala has a brother named Mheetu, as well as the fact that the books indicate that Mufasa and Scar's ages are too close for them to have been born in different litters.

I believe that Simba may have been born towards the end of a drought, and that stress from the lack of food may have led to his litter mates being stillborn, and making his mother infertile (which I say mainly because Scar is exactly the kind of monster to force his brother's wife to have his kids, and none of the outsiders look suspiciously like simba.)

This is also part of the reason why previous generations of royal cubs didn't get the same kind of treatment, as everything I've read seems to indicate that Scar and Mufasa were kind of allowed to do as they pleased so long as they never left the pridelands