All of the Ivies attract students who are fine musicians, as do most competitive colleges. At many fine schools, you can double major in music and another field (this can be difficult if the other field is science), minor in music, or just become involved by taking lessons and/or music classes and joining ensembles. (Actually, that’s probably true for not-so-fine schools as well. ;)) Or, as you suggest, you can simply work independently, although my suspicion is that it doesn’t end up working out that well for most people.
The standard advice I’ve heard is that if you can even consider working in a field other than music, go for that and keep the music for enjoyment. There are a surprising number of accomplished adult amateur musicians, and a good pianist is always in demand. 