TL;DR
Saving and immediately stopping the simulation in a multiplayer game would leave all players with exactly the same save file. This save could then be used to reload the game at the exact same point for everyone, removing the need to download the map.What ?
It would require another kind of saving mechanics. The server pauses, saves, make all the player save, and immediately eject everyone from the game.When starting a multiplayer game, the server loads the file and waits for other people to join with the same file. Checksums are exchanged to ensure that everyone has the exact same file. At some point the server decides to start the game, and everyone is immediately in the same game without any download. It's just a matter of loading everbody's local copy. It would also remove the catching up.
From a GUI perspective, it would need some "Save & Stop" in the in-game menu, and something like "Host synced saves" in the main menu.
Why ?
As maps get bigger, it gets more and more tedious to join a game with a weak internet connection. When playing with the same people, always at the same time (like a normal coop game), always downloading the same maps seems unnecessary.(Thanks to all the team for making this really nice game)