To me, broadening participation in computing means to get more people interested and involved in the computing fields. I think that there has been a serious lack in promotions for computer science and other related fields. It is a great idea to get outreach to younger students and get them exposed to computing and the "cool/fun" things it can do.
I think the SPARCS Outreach definitely does this by going to the middle school and getting the children interested in different ideas of computing. We have a diverse group of children who come to the sessions, and they usually have a friends there with them. So I think that having these sessions spread computing to kids who might not know about it otherwise. We try to keep the activities fun and interesting so the students stay intrigued by it.
If I were exposed to BPC outreach in k12 I'm not quite sure how I would have changed. With absolutely no exposure, and actually disincentives to do computer science, I still did it. My whole computer science career decision was based off of my laziness in geometry and one online computer class. I already had other extra curricular activities that kept me busy, so I probably wouldn't have joined the club. Younger students on the other hand would benefit from the program I believe, because they haven't had the opportunity to get involved/ interested in other options that seem 'cooler'. I have to agree that a big opponent to BPC is the geeky stereotypes, so if we get to kids before they start to worry about their 'image' it might work out. Then eventually the demographics of computing could change.