I can take a look at this game if you'd like. Just let me know the ID

Unfortunately, GMR may actually prevent that from happening. It's built in as a safeguard, because once all players but one are AI, Civilization will treat the save file like a single player game, rather than hot seat. This means that it'll never display the hot seat screen after you finish your turn. Instead, it just processes everyone else's turns then starts you on your next one. Under ordinary circumstances, this would be unintended behavior, so we have safeguards to try and prevent it from happening.

Sorry about that! Our server clock gets out of sync when we update the server, which Steam doesn't like. I just corrected it, so everything should be good now.

Hmm, you know, I'm not really sure. The map would be embedded into the save file, but the thing is that a Civ V save file also includes what mods are in use for that save. So, even if the map embeds, the game may still think it requires a mod to load.

In the past, I've experimented with turning a standard multiplayer game into a hotseat game, but it doesn't work. So unfortunately, there's no alternative way of starting a game.

However, you should be able to get through your initial turns fairly quickly if you go to your settings in the desktop app and turn off automatic Civ closing. That way, you can just hop back to the main menu after you take your turn and not have to restart Civ every time.