I think it's a good idea in some aspects, but not a good idea in other aspects. I probably won't participate myself.
I don't want to be 'forced' to go for a certain victory condition, so get rid of that idea, IMO. Some players decide when the game starts (or before) what kind of victory they want to shoot for, or what interests them. And some, (like Zachriel) decides as the game progresses. Some players have become really good at certain victories and they enjoy these games more than other victory conditions. If they are in the mood to try something different, they will do so when they feel like it. A seasoned spaceship player may not want to try and get a quick conquest, and a heavy warmonger may not want to try a diplomatic victory. Sure, they should probably try something different once, just to 'broaden their game experience', but having them try something different when they are also under the pressure of letting their team down if they don't play well enough, doesn't make for much fun. And if they want to try something different, they can always do it away from the GOTM, in their own private games, whenever they get the urge to do so.
Me, I wouldn't be a great leader if I was trying to hand out advice for a quick spaceship victory. Sure, I know some basic tips from reading the forums, but to get the best advice would be to ask the players who got fast spaceship victories in the past (Kemal, Dave McW, Ribannah, etc.). Some players are better at warmongering tactics early in the game, and others are better at fast tech speed, and others are better at building culture, and others are better at diplomacy/trading.
Now, I know someone will probably chime in with some silly comment like "the best players are very good at all the above mentioned things", but that isn't my point, the point is what specific thing they are the best at (call it their 'specialty' if you want), and what they ENJOY doing.
ICS players and players who build cities further apart won't mix very well. Many players who don't use ICS, have never even used it (or used it wrongly), and stereotype it and don't really understand the full process of how it works (what to build, terrain improvements, etc.), so they would be probably giving the wrong advice to an ICS player. The same would apply for someone that like to build up a tech lead themselves, and another person who likes to do 'pointy stick' research.
How often would 'teams' be assigned? Can players drop in/out of them on a month-month basis? Personal commitments and other things can affect whether a player can play a certain month or not. Some players just don't like playing on certain maps. Some really hate playing on larger maps (time involved, computer speed, etc.), and some hate small maps. And after a recent experience, I have further evidence of my hatred for island maps.
I do like the idea of the better players taking a look at someone's game and offer advice, but how to do that is debatable, when you got spoilers to worry about and what is considered to be the best time to give them advice (during the game could cause spoiler problems, after the game could fail to have the tips/strategies sink into the player's mind). 90%+ of the tips are situational dependent and applies to one game, but not the next.
How about non-competitive teams (we won't compare team scores) that form based on victory condition they are aiming for. Various threads will form for each of the victory conditions, and players can discuss with others going for the same type of victory about the progress in their game, and from that thread a person will emerge as a leader (they may not be directly indicated as the leader, but as information is shared, most will get a feel for whose advice they want to follow).
For example: In the general discussion forums, and strategy forums, every once in a while someone posts a save and asks people to 'save his butt', because he doesn't know what he is doing wrong, or why the AI is beating him. I may look at the save and see a few things, like the number of workers or terrain improvements, then offer a couple of ideas that they could do. someone else takes a look at it and sees a few more things, and offers a couple of different options the player can take. With one leader, the new player is likely to just follow the leader's advice and put 'blinders' on, thinking that is the ONLY sollution. With multiple options given from a variety of skilled players, the player takes a look at all those possibilites and decides what is the best move for him and his play style, and becomes a better player instead of just a clone of the leader. Sometimes, the less skilled player has better ideas for a given situation than the higher skilled player. The higher skilled player is only better because he is better in other aspects of the game.