It shouldn't be restricted to the party leader, nor should there be a several minute cooldown, because both of those interfere with the proper, useful purpose of the ready check:
You approach a boss fight area and the tank initiates a ready check to see whether to begin it. Everyone except the BLM is ready but he needs a moment and says no. A minute later the BLM is ready so he initiates another ready check himself, which tells the whole group that everyone is now ready to go.

It should, however, be disabled during combat. That should stop the worst of the abuse of it, and without interfering with the feature's useful purpose.