It seems to me that this is mostly Putin wanting to appear to be a savy problem solver and taking a gamble at being the first to the table, figuring that the other 3 in the quartet will back him up if things turn into a mess.
If I were a "conspiracy theorist" type of guy, I'd actually start to wonder if he wasn't "pushed" into making that move. Putin can get away with it at home, Britain and the US cannot, and nobody else would want France to take up that role....