I sometimes use KMC chains. The pins have shaped ends that help to hold the chain together. When you drive a pin out with a chain tool, the ends deform and create a weak link. You really should use the missing link at the place where you broke the chain.
You're right that it isn't a true master link. They aren't made to come apart again. I have a couple of spare Missing Links. I keep one in my toolbox and another in my seat bag.
I like to use the Missing Link with Shimano chains, too. I have a deep and abiding hatred of Shimano's special rivets.
__________________
Religion is a good thing for good people and a bad thing for bad people. --H. Richard Niebuhr