Depends on whether you want to be a technical engineer or more of a manager, I think. I am sure there are Harvard grads at all the top chemical companies, but they probably aren’t all doing technical work.
MIT is #1 chem engineering school, if you can get in and want to go full technical maybe pursue a PhD.
Also, why chem E ? Do you actually want to work in the chemical industry or do you like chemistry or is it for potential for jobs ? Chem E is often process engineering …