"Whenever I look at Aoi-San, why do I feel like I have to take care of him? I wonder if I feel that way because he seems like someone with deep wounds."
Well you could feel that way because he's hopeless at taking care of himself.
Now Minazuki is being super nice by helping Aoi out. That's fine as friends, but IRL you would very much not want to fall for someone broken. Aoi has been majorly depressed for ten years. Would he be able to pull right out of that, even with Minazuki's help? Not likely. Now in this story, I'm guessing everything will turn out okay even if they catch feelings for each other, and Aoi will be able to start functioning in society. Realistically though, if you find a romantic partner, it's much more of a recipe for success if you team up with someone who is more or less equally as emotionally healthy and competent as yourself.