I guess I'm adding personal experience here, but also Joyce was not able to totally care for herself pre and post brain surgery. Buffy was attending doctor's appointments, hospital visits, taking care of the regular chores around the house, babysitting Dawn, picking her up from school, on top of her dutiess as the slayer. She moved back into her home to try to continue to go to classes but be there to take care of her family, including both protecting Dawn from Glory and her mother's health issues as reasons for giving up her corner dorm. It is seen in multiple episodes from Family through Into the Woods how Buffy has stepped up to be caregiver for both her mother and sister. And while these are a span of 4-5 episodes, it is really a span of a couple of months, as Family was set pretty early in the semester, even though it aired in November. Season 5's timeline is a little wonky because so much happens before they go back to school, as evidenced by Buffy stating Dawn has been in existence for 6 months, putting the events of Buffy v Dracula six months before January. So, Buffy had been caregiving for Joyce and Dawn for a good while before Buffy's birthday.