Really? imo, not liking the Columbia Core makes it a no-brainer to not apply! Don’t forget that the point is not getting in but having a good college experience that sets you up for the next part of your life.
You say you don’t know what you want to study, so you seem to be substituting famous names for working harder to figure out what will work best for you best. You won’t be applying for almost a year, so you have time to keep thinking, keep investigating different college environments, looking at what a BME course or a business course really means: the classes you take, the pathways it opens. You have time to think about what you take to natively- classes and activities that you genuinely enjoy and find satisfyingly challenging. Keep honing, refining. Your applications will be stronger for it, your chances of success better, and your odds of being happy with your ultimate choice higher.