There are definitely types of game that will and won't work for this. I have played this way, and very successfully, using an episodic game (we used Prime Time, but there are plenty out there) where the continuity was less important. Here are the things that are important to make this work:
Each Session (or GM-period, if you choose to do 2 or 3 games before switching) needs to be self-contained, with its own start, middle, end, and a complete story told.
If you want a longer-running story arc through these episodes, then the GM's have to work together to build it, or one GM can run the first and last games, and give each other GM things to insert. This works if everyone's mature about it. Games that share narrative responsibility are great for this, as the "Arc GM" can insert things as a player, when they get to narrate. (FATE, Prime Time, Dogs of the Vineyard, etc)
As far as swapping out players goes, there's multiple ways to handle this. If your game is an episodic crime-drama for example (think CSI), then the team comes back together at the station at the end of each episode, so one person not being there next episode is easily explainable.
Another way to handle it is for each character to have a separate skillset, and for a GM just not to write anything that concentrates on his own player's skills. For example, if I was playing the blood-spatter guy (Dexter inspiration), then the episode I run would not have a live crime scene (the body washed up on the beach, or there IS no body, just a missing person, or something else similar). Then, although my character was "there" the fact that nobody ever talks to him for an episode is fine. In Shadowrun, if I was playing the Decker, we could have an adventure where all the hacking was done from base, so there is no need to take the decker with us, then he would be involved and useful, without needing to be as present as the PCs playing this game. This is the only way to make it work if you can't negotiate a single-episode-and-back-to-base type game, as characters can slip in and out of the shadows as needed.
I certainly would NOT suggest the GM plays a full character as part of the party. As you've said, that spells doom...
I've also played games where the PLAYERS all knew all the secrets, and we had to separate player/character knowledge. This meant that players were deciding when their characters would make the right decision vs the wrong one, based on what would make a better story. Again, with a mature group, this can work very well. If you can handle this type of game, you may not NEED a GM. If everyone works out the story together first (we find a body, these are the clues, these are the possible suspects and avenues for investigation, and this is the killer in the end) then play it together, there's no need for a single narrator. You could even randomise the "who the killer is" part (or "where the treasure is" or whatever suits your game, I'm sticking with my police drama for examples). Then nobody would know which avenue to follow first. It could lead to an anticlimactic game though, if you go straight for the goal without any of the conflict...
SO, there are many ways to handle this, all depending on what your group can and can't manage. Of course, there are some groups that just can't do this at all, and you may be better just accepting that your "campaigns" are going to be made up of 3-session adventures, and that's ok too. You could always cycle campaigns, and come back to each one for another adventure when the GM has un-burned out.