Taking command of the bridge during a crisis is not the same as assuming permanent command of the vessel.
No, but it's not a stretch to see how such an incident resulted in his eventually getting the vessel. What can be inferred about the event is that Picard was the highest ranking officer on the bridge at the time the captain died. Assuming command of the vessel was the appropriate thing to do. He may not have earned permanent command right away because of his rank, but his actions would surely have resulted in a commendation or promotion, putting him on the fast-track for the center seat.