If you’re going to play with the idea, then write the essay. At worst, it will be cathartic, and then you can put it away and start your actual essay.
IF you have any shot at making this work, then I would suggest a very brief intro to your dad’s problems. This essay should not be about him. It’s your one shot at separating who YOU are from your statistics, and you have only 650 words with which to accomplish that. So a brief intro to dad’s illness, then an essay about how it shaped you into the type of person they want on their campus.