Just spitballing, but I would guess that the local or national government has responsibility for the island, and when they knew there was someone very ill on it, they were compelled to treat him. And with that history, they couldn’t send him back. It would be best if they could take him back and send a doctor around once in a while instead of taking him prisoner, if the legal and social framework could stand it. Given the nature of Japanese bureaucracy, I doubt that’s in the cards.
Edited to add:
Totally still timely and apropos of this situation is Ikiru (To Live), one of Kurusawa’s most affecting films. It’s about a bureaucrat who has a short time to live, and is frustrated in carrying out his simple vision by the bureaucracy.