I don't have the game open, but for each delivery mission, after picking it, you need to transfer the cargo. At the other end you need to deliver the cargo and see the mission effect.
So for 6 missions: pick 6 missions, transfer 6 times, travel, deliver (transfer) 6 times, see the effects 6 times.
With the new layout, once improved, you would be able to pick 6 missions, transfer the cargo 1 time from depot, travel, deliver (transfer) 1 time from the depot and hopefully see the effects 1 time from the combined influence screen.
So I can see the benefit, just not quite there yet. Think the influence screen would make more sense when you leave the missions area, so a recap of all completed missions once you are done. No need to give an update after every mission is handed in.