Question: Is it your intent within this system that the normal Aid Another action can be taken? When the enemy is time and not failures it seems like frivolous or cowardly Aid Another would be discouraged but if a party really thinks that it's the better strategy for them, it seems fine with me as long as they realize that's giving up a chance for a success.
Thought: If you don't want just 3 segments, but also don't want unlimited time where they just take damage, lose healing surges, or something else with time, I wonder if it's viable to increase the number of successes necessary for full/partial success. E.g. maybe the king will listen for more than 3 segments but if you want to take more of his time you better come up with something good fast. It's kind of like saying "Best 2 out of 3?" after losing a coin toss. If there's an expected # of successes per segment on average that info could be used to decide how much to raise the threshold for success if you allow a 4th or more segments.