If you’re in a public or private school, and you don’t fulfill a minimum of your academic requirements and attendance for the year because of sickness, you are always going to be either held back for the next year or expelled. You are never going to get to graduate with your year. It cannot legally work that way in public schools.
Some private schools make up for lost school with letting the kids make up homework in a rush and testing them, but if you’re still sick, that kind of exhausting catch-up isn’t going to work, either. (It’s usually done by athletes who are injured severely, but still have lots of go-go-go as they recover.) But they are also bound by state laws about schooling, and there’s only so much flexibility they can legally give.
However, there is a way to fulfill a sick kid’s attendance and academic requirements, and stay good with the state and the school.
It’s relatively easy to get a home or hospital tutor who’s certified by the local district, and the school or district is usually more than cooperative about it. In pretty much all public schools, the district pays for the tutoring, and the tutor works closely with the child’s teachers. Private schools usually will make some kind of arrangement with the local school district or with tutors they know.
I know this, because my mom worked as just such a “home instructor” during my entire childhood. She was a fully certified teacher, working part time to stay home with us kids, and making most of her money from the district. (She was also available as a freelance tutor for healthy kids who just needed help with their schoolwork, but in those cases the parents paid.) Even very sick and dying kids could keep up with their schoolwork by having an hour or two of individual instruction every week, and then doing homework a little each day. Usually kids ended up ahead of the rest of their class.
Also, parents are always informed about the rules of being sick for long periods and away from school, quite early in the process. Schools have to do this routinely as part of managing attendance. Instruction for sick kids is not so much offered as made difficult to avoid. Like I say, this is a legal requirement from the state board of education. There is state and local paperwork for the school to fill out. No school is going to encourage someone to skip it merrily so that the school gets fined or has funding cut.
If the kid was too sick even to receive tutoring, then it might be reasonable to ask for a refund from the private school. But other than that, I don’t see the problem.