Ah something I am an expert on. My agency has jurisdiction over 472 named islands of those islands only 205 are public lands or are generally accessible to most everything. An island here would easily fall in the 1.2 mil and up price range depending mostly on how developed it is.
Hospitals? we don't need no stinking hospitals. We have afterhours medical care but if it's bad it better not be so bad you can't live for more than 25 minutes, because in the best conditions that's how long it takes to get Airlift (helicopter with advanced life support and a E.R. Trauma nurse) to your private getaway.
Most of the islands don't need upkeep other than routine maintenance. There are a handful of businesses that will do it (usually doesn't cost more than 30K per year) and most land owners just hire someone to take care of things as needed.
Biggest problem is accessibility. Due to shoreline management not all the islands are allowed to have a dock or airstrip. Most owners will come in via chopper and can get away with a helipad/field without needing permits and such. Oh and cell phones don't work in the most populated parts of the islands let alone the little 4 acre islands floating around out there.
I used to work part time for a large (180 acres)estate owner on a private island. It was an interesting job and provided some incredible experiences. I know that I would not trade the life I have now for what he doesn't have.
In fact, I did not meet a single person who had a mountain of money that I would trade places with. I'll gladly wonder how I'm going to pay the water bill while playing on the beach with my kids. As opposed to worrying who is going to kidnap my kids on the beach for ransom money. Icky way to have to live.