Stray Animals

Animal shelters often have to kill animals because they were considered "stray pets".  I know that this might be considered a horrible idea, but why don't zoos use those animals that are about to be killed to feed zoo animals?  I know that for some animals are fed live animals, such as mice or rats.  Wouldn't large cats, hyenas, wolves, etc. prefer a fresh kill, which allows these animals a chance to respond to their natural instincts? 

I saw something recently about obesity in zoo animals because they don't get enough exercise.  One zoo (possibly more) is weighing the amount of food for each animal.  If  a large cat has enough room, wouldn't it be more beneficial for the animal to chase and attack a meal instead of throwing hunks of meat?