Would they be mandated to give out the server code that people could run their own servers?

Sort of. The Idea is that people should be able to run their own servers, but developers wouldn't need to give out their code. All you need is the server binary. After all server software is just that software, just like the client and they don't need to give out the source code for that for you to run the game. Alternatively they could patch the game so it's peer-to-peer. (and yes in this case that would be unreasonable as the game is not successful enough to even break even)

The initiative is so ambiguous (to the extend that it is - I'd argue that it's a lot clearer than many people claim) because it's not actually legal text. It's not supposed to be. All it should do is describe the problem and explain why the problem falls under EU jurisdiction. Everything else is supposed to be handled by EU lawmakers after the initiative has met it's signature goal.