It wasn't an argument. You've been doing not so well for a long time and your mother has tried to help you possibly in the best way she could think of before suggesting taking you to get some professional help. Depression is a dark, moist pit. I know. And I know it can get better but it can also get a lot worse if you drag people down because you'll then start to guilt trip yourself into having caused misery beyond your own existence. Do yourself a favour: gather all the negative emotions you have and face them. You've already done it by going on a 2h monologue about how awful and meaningless it all is so this part should be easy. Now, think hard if you want to change this situation. If you do, then think hard about ways to do it. If you can't see a way out or you find it hard to commit to it, ask for help and/or support. You're 16 so there are a lot of wisdom nuggets lying around waiting for you to pick them up but from what I see you're a good kid. Feeling bad about causing hardships for your mother and taking the initiative stepping up to cook and feed your siblings when it's required of you. You're a caring person and that's precious. Here's today's nugget for you: take care of yourself as you would do for a dear friend. Don't think people will deny you help, or that they don't want to aid you if you need it. Many people are really eager to help others if you so much as hint at it. It makes us feel good so you don't even have to feel embarrassed or think you're taking advantage of our kindness. If you had a friend in your situation you would like them to tell you so you could help them out I bet :)

I don't like telling people what to do but I love giving advice when I think someone's asking for an opinion (and a reddit post like this clearly is imho), so excuse me if I have been a bit pedagogical (and for the wall of text lol)