It's 100% up to the Planning dude on duty in MSP. He has the "big screen" and tracks the downline implications of holding the flight.
The root problem is that very few (read: "none") of the agents will stick their heads above the ridgeline on this one. They are tracked for their ability to get flights out before scheduled departure time. (The statistic is posted daily on our company website). If a particular agent isn't authorized to hold a flight, they get dinged for the late departure, and it can lead to discipline if their batting average drops too low.
In the "good ol' days" the captain could clue the gate agent in on the flight time for the segment, and delay the flight to connect all the pax. Now, captains are getting calls from the Chief Pilots if they push late after the door was closed early by the agent.
I make it a point to check with the agent before each of my flights for the "big three": close connnections, non-revs, and jumpseaters. I always let the agent know I can help get them all on. So far, the only invented excuse I've had to use more than once has been an "intemittant headset mic". It doesn't show up on the ECAM, and can be easily swapped-out or re-seated in the time it takes to get the runners from G14 onboard....or the non-revs loaded. The last time the Chief Pilot called I told him the truth and my cover story. He thanked me and said he was going to save my story in case someone else couldn't come up with a good one on their own when he had to call them.
Crazy, huh?