The inherent gift is demonstrated in game play through spells. Spell that come from within. Innate spellcasting, right?
Your description is really good. Makes me think this idea is more like a superhero than any other class. Born with special powers, exposure to magical "radiation", etc. I wonder if that is why it gets so much pushback from some players?
Seems like the subclasses should be more themed in restricting spells, focusing on the gift the sorcerer has.
Maybe just go full on superhero. Not spellcaster at all. They wouldn't "cast spells", just use there spell like powers. Something like invocations might be more appropriate.
It would be fun to do that. Make it part of the game. If Warlocks bargained for magic from different sources instead just having one patron.
An elemental sorcerer then could cover fire, cold, as fire, ice.
Good discussion.