I'm upvoting for the edit. I hope you reflect and figure out how to better address the issues here and that you rekindle with your friend.

I don't think you're evil but you were definitely an asshole. As a friend, you should try to be understanding of your friend's situation and if you want to have a closer bond, that means putting aside some elements of your game for the serious stuff. If you've been friends for years and care about these people, there will always be more campaigns, no matter how good or bad this finale might go. I also think it could help to think of less extreme scenarios. If you're someone who has a lot of stake/care in these campaigns and can get upset easily by the idea of it being ruined? You could suggest to play a one-shot with the sister instead. The biggest issue is thay you refused to see any of the difficulties your friend was facing and acted very entitled and difficult over a situation that did not need to be blown out of proportion. Regardless of how much you like to keep these sessions 'perfect', there will always be issues to be faced in real life that may interfere with scheduling, or audience, or host. You will have to figure out to deal with these things without it turning into an argument.

I also hope that you learn to be more considerate to the issues of your friend's little sister. Reflect on when you were 11. Can you imagine what it would be like if you didn't have your mom at that time? Maybe you can. If you can then it should be obvious that an 11 year old is allowed to be clingy, especially under those circumstances. It doesn't help that you placed her as a burden on you. I don't think your friend would take kindly to that view, nor would most people. She is not a burden and she's allowed to exist in the space that she does and her existence shouldn't ruin things for you. Especially not so badly to cause this fight.

Also, I'm not sure how many people addressed this but you referred to hanging out with him and his little sister as "free babysitting". I already said that you making her a burden was bad, but adding to that. You being singularly annoyed by this and expecting your friend, who will already be hosting you and was dealing with his girlfriend's mother in the hospital to also pay for a babysitter for YOUR specific comfort is selfish. If it bothered no one else, you were angry and expecting him to pay for a babysitter and to make his niece uncomfortable for your comfort. She is also an 11 year old child that is his responsibility to take care of and make sure is comfortable. You are his adult friend. He is also hosting this game but has his own life outside of that. You've stated you can't host and felt it unfair you were told to host if you didn't like it. The thing is that the host gets to make calls about their own living space as that is their living space. You do not get to have someone offer to host for you and then demand things be done your way in their home.

Anyways, I really hope you take the constructive comments to heart. I don't like bullying but I think this a sensitive issue and that people are prone to being a bit hot blooded when you're behavior is so out of line.