Honestly, this is how I thought it would be. Each week you would get an Elite Mark in your Journal as a quest, you form a party, you go to the area designated, initiate the quest, and kill it. Only people in your current party can attack your mark, so you don't have to worry about anyone claiming it before you.
I know people wouldn't be able to farm endless Allied Seals this way, since the game would only technically allow you to spawn one Elite Mark per week, but maybe they could have it so that if you complete the quest for five Regular Marks, you would gain access to another quest for an Elite Mark? As it stands, very few people [if any] do the quests for Standard Marks, since their payout is so low and likely isn't worth their time traveling the distances to kill it.
To be best-in-slot, of course, and make our clears even faster, not to mention being better prepared for the Third Coil of Bahamut [when it's released]. Also, depending on if we get bored enough, it'd be nice to have just in-case we feel the need to try Brutal Coil. I have a feeling we'll need all the help we can get in clearing that...