The romance of the early part of the last century has, unfortunately, disappeared.
Palmyra is a great example. It looks like no one owns it and you could just roll up and drop the pick, but thats just not true. Its a wildlife refuge and until recently any visits were illegal... But this has changed
Limited visits to the refuge are allowed, including by private recreational sailboat or motorboat. Visits must have prior approval, with access to Cooper Island arranged through the Nature Conservancy.
Every other island in the world is sovereign territory for some country and you are required to live under their laws. This means most of the Pacific is owned by the french and you get your standard 6 months and then get the hell out of there. Some of the smaller island nations are much more draconian.
There is also a practical reason why so many islands are not inhabited. Their aint no anchorage, nor access to their lagoons. You physically just cant get on them.
Any island big enough for fresh water and accessible by sea will have people on it, or be owned by a family etc
Then you have the cyclone season...
There must be places you can go and the finding of them would be an exciting adventure, but it would be time consuming and full of unforeseen difficulties.
Have fun researching it, and finding your paradise.... But when you do find it do not tell anyone because the next day you might find a few boats beating into the anchorage.