Yes, BJJ is both Self Defense, and Sport.
At my club, we train both. We always have a self defense portion of the class. I think a good solution to the problem you suggest is to have different classes scheduled for each. Self Defense class should be taught on a seperate day at a seperate time. This way if you are only in BJJ for the sport side, you can train that. If you want to train BJJ for the street side of it, you can train that.
Though this may still diminish the whole self defense aspect of Jiu Jitsu, because if students are not forced to learn this as part of the class they attend, and no one wants to train the self defense aspect of it, then it would probably be dropped from the schedule.
My argument is for those who keep saying this is not for the street, and that is not for the street. My point is that, what we learn can be applied to the street fighting scenario, and will help you.