Private equity is very much a "who you know" type of industry to break into. Your best shot based on the schools listed are Stanford and Chicago.
All schools do in fact boast classes that focus on the softer skill set. If you're interested in what recruiters think about how well these schools teach the soft skills, I'd recommend dropping by a book store one day and reading through the WSJ's guide to rankings for these specific schools. While I think the ranking is somewhat flawed, the anecdotal insights are often pretty good. Just don't forget, there is no such thing as unbiased (even with recruiters).
All of these schools have a general management curricullum, so to the degree that you want to learn soft skills, you just sign up for those classes. All of the schools mentioned have made strides to their focus on leadership in the last 5-10 years.