The wristbands the kids wear in the club have their stateroom number AND their lifeboat station shown on them. If there is an emergency that would require you to go to the lifeboats the CMs will get your children to the right lifeboat station. In this situation it would be more dangerous and MUCH more time consuming for all the parents to come to the Club or Lab to get their children. The parents should go directly to their lifeboat station so they can take charge of their children when the CMs bring them.
One advantage the CMs have is that they know the ship VERY well and know the shortest way to get places using crew stairs, etc that would be less crowded. Since the Club/Lab/Flounders are on Deck 5 and the lifeboat stations are on Deck 4, your children are often closer to the lifeboat stations than you are.
My DD is a CM in the Club and she says if an emergency is announced during Club hours, EVERY CM who works in the Club has orders to get there ASAP to assist in evacuation procedures regardless of where they are on the ship. They don't make a big deal about it to the guests, but they do have defined procedures, all crewmembers have defined emergency assignments AND they practice them regularly.
Hope this helps put your minds at ease. If you have other specific questions, please feel free to PM me.