As for Harvard, you can absolutely go to Harvard Medical School to become a pediatrician, if you chose to apply there and get accepted. I would also like to remind you that there are many excellent medical schools which will also give you a wonderful medical education. Teaching methods vary at different medical schools and you may find yourself interested in a different education experience. Medical schools also give you a general education, not a specific pediatric education.
So, you could choose to go to a different medical school, then return to the Harvard system for your residency, which would be at Boston Children's. That is where the specific pediatric education would happen.
Harvard certainly has some benefits, but it doesn't give you extra credit on your general education. As an extreme example, even if you went to Harvard for medical school, if you failed your USMLE exam(the final medical school exams) , you still wouldn't be accepted into residency programs. Harvard would give you some connections, and you would have the opportunity to be at Boston Children's Hosptial as a student, which would be a whirlwind of an experience. On the other hand, being a part of large programs like this tend to mean you are part of a gaggle of students and get less one-on-one time with mentors and attendings. It's all about balance and the experience you want.
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