From what you've written, I would agree that the center is partly to blame for what happened. Firing you probably placated the angry parent and relieved the center's director and remaining staff from having to accept blame for his "missing" child. It probably kept the parent from calling licensing to report the incident. "Phew! That was a close one!" for the center.
The fact that you knew there were 7 kids outside playing, only counted 6 when you went inside and didn't pick up on the fact that there was one missing makes it your mistake, too. (I know you know that.) You're human. You made an unfortunate mistake. I'm sure you feel a lot of guilt over it and probably a lot of anger and frustration over the fact that the center blamed you and basically used you as a scapegoat. Two years is an awfully long time to be beating yourself up for what happened, though - especially since the child was found and was okay. Hopefully, venting about it has helped. If not and you find you can't get past it, you may want to try talking to a professional about it. You shouldn't have to continuing feeling like a "monster". It's time to let it go.