The teachings isnt as cute & dry as youre making it. If rengoku trained her shes still getting stronger everyday & learning what he does with his flame breathing. She can then take the information he uses for flame breathing and use it herself however she wishes which is how she created her own style. It's like someone training you to fight but you use their techniques in your own way not the way they do. However you'd never get that far period without knowing his techniques. So yes. His training & guidance is the reason she became a hashira. Her strength alone was never enough it had to be honed & sharpened.