Absolutely not! In fact, it’s really common for people start to do an activity because of external rewards or punishments, only to grow to believe in it. Like the child who only does violin practice because their parents insist upon it, who complains as they scratch out horrible sounds and tend their sore fingers. But after a while, this child (may) go on to love playing the violin, practising for hours and forgoing other pleasures to attend orchestra practices. Violin playing started as externally motivated, and transitioned to intrinsically motivated. But in order to make that transition, the three needs of autonomy, competence, and connection had to be in place.
So in this case, that was probably a supportive teacher, who notices and comments positively on progress made, who allows freedom and creativity, and parents who allow them to sometimes skip violin practice as time goes on, or even make it known that they can quit if they wish. Also, the theorists do recognise that the range of behaviours which can become self-regulated increases with age. A three-year-old cannot truly internalise the idea that they are an environmental steward to the world and therefore choose ethically made long-lasting toys. But they might choose a wooden toy over plastic if prompted to by a caregiver they love and trust, and they might want to learn how to fix what’s broken, rather than discarding it, if they’ve seen their parent fix broken things before. At this age, the child is motivated by connecting with their caregiver, not the values the caregiver holds.
But over time, with explanations, discussion, autonomy, and competence, the value of environmental stewardship may become internalised.