Does how I have behaved make me a bad person?

I think she needs help. You are probably right about the depression, and getting treatment for that would be a start. If she can do things when she has to perhaps she would respond to a request like "Could you please take the children to school today" The other thing is to include her is some decision making. Make her think. "I want to do something at the weekend which will be a real treat for us all and something we will remember. Any ideas?". The other thing is her weight might make her hate to go out in public. The doctor might be able to help with that too. I think your first step is for her to have a thorough health check and then go from there. It is one thing being supportive but it is not kind to let her let herself go completely. And no harm done if you had a vent of how you are feeling. You are still there aren't you? And concerned about everything.