From my own observations, deductions, things I’ve heard, and other pieces of evidence, I’ve come to this conclusion: Brian Burke’s only job in Calgary is to say things.
I came to this primarily because of his job title, President of Hockey Operations. Think of the teams that have POHOs (or similar titles): Edmonton, Vancouver, Toronto, Columbus, and Calgary. All teams that have been rebuilding in recent years. Think of the people hired to be POHOs: Kevin Lowe, Trevor Linden, Brendan Shanahan, John Davidson, and Burke, respectively. They’re all huge names, well respected in the hockey world.
This is mostly because, in my opinion, POHO is mostly a symbolic position without any real responsibilities.