They certainly hold some power, at the very least as you pointed out just by being probably closely connected to newsoutlets due to their buisnesses and charitable functions, so when they have something to say, they can get it "out there"
Unless something sliped past me, Burke being the Dean is something that many people confuse, but he's not. While we only know that his professorship is honorary, his position in the College seems to be purely administrative. Maybe head of funding, maybe head of accounting. To become the dean of a college "honorary" won't do, you need some high education in leadership, or teaching or education.