Actually i do not think, that "Quick Start" was ever implemented in FreeSpace.
It was an intended feature for Multiplayer in FreeSpace: The Great War to start a quick multiplayer match. Without Server Browser etc.. Simply Click and Join the next available match. But that was never used even in the first Freespace and so it is a left over in the code.
No idea, why that one got into the community created Soyakaze Mainhall... or why that one is active in Single Player.
Yes, of course. Today it could be implemented to use to join a random multiplayer match or even to start a random single player mission. But afaik we are not there, yet.
Quick start has nothing to do with the briefing room button, you have to use to continue a campaign, however