The rest idea works well with the time limit of some of people you have to save: you can take a deep breath and recover, but that means less time to save them, what gives some depth to the game - you can go through it just trying to see the end, what's more about having enough persistence
, or trying to save everybody, which forces the player to learn how to be fast and more precise in each stage.
Maybe some sort of medicine in the stage that gives you temporary powers? Some item that does the opposite would be a nice addition, and it's not that used in games. Examples are the red Continues in Jurassic Park for Master System, and the Eggman Monitors in Sonic 3:
http://sonic.wikia.com/wiki/Eggman_Mark