Nation RPG

Playing At The World by Jon Petersson is a fantastic book on the history of D&D that I take the opportunity to mention whenever I can. If you are interested in the history of games (and imo you should be if you are interested in designing games) that book is a must.

Diplomacy comes up a lot in that book. That game of course does not have any roleplaying in it by design, but what I learned from the book was that groups of players started evolving RPG-like playing styles for that game. Instead of just playing the game as a boardgame by the rulebook they started playing "in character" with some groups having someone even print fanzines with articles about the made-up events going on in the world between the various nations. And one active player on that scene in the 1960's was Gary Gygax. So roleplaying with nations sort of happened before roleplaying with fantasy heroes. :)

You might also want to look up Matrix Games. Not the company (that publishes computer war-games) but a type of political role-playing games. I do not know if they are ever played without a GM, but then I do not know much about that obscure genre of games at all. Can not even find a page about them on wikipedia and they are not trivial to google, but here are two pages that came up:

http://www.cbc.ca/news/politics/isis-military-gamification-national-defe...

https://www.rt.com/news/346544-isis-canada-military-game/

Of course after that you could go down the rabbit-hole of government-run roleplaying/wargames (and the mega-games scene), but I think they typically have one or more umpires employed to organise things, rather than just having each nation run itself and follow some rules for interacting with the other nations.

This definitely sounds like a fun idea. I am worried it is something that would require far too much time for me to get into personally though (same reason I avoid any regular RPG, other than ones I can play with my kids, but even that does not happen very often).

Quick thoughts:
1. With stats, try to make them more volatile than standard attributes (using D&D as an example, STR, WIS, etc.) and less volatile than other numbers (experience, gold, hit points). Some may be linked to limit or boost others or perhaps surprisingly unrelated. (Think of poverty levels of major nuclear powers)

2. The two issues you will probably find yourself addressing at some point are time scale and the interdependence of countries. Contemporary earth may be a poor model for a good game unless you exclude the economic and nuclear superpowers.

3. The idea that you're thinking GM less is good--standard practices, etc. should inform your game, but to pull this off, you're going to need to be open to anything.

4. With the above in mind, rather than starting with rules, start envisioning and describing (preferably on paper) sample play sessions and see where that takes you.