i'm sure a lot of the islands are part of nature reserves, but also a lot of them are already privately owned with a house or dozen on them, especially around stockholm/gothenburg.

You can totally find some island somewhere to purchase and live on, it's just tedious and the island will probably be located in bumfuck nowhere and now you suddenly need both a boat and a car to get anywhere, and it takes 2 hours to get to the closest grocery store..

as for allemansrätten, if you have a tiny island with a house on it that is honestly about the only way to nullify the right to roam, since you're not allowed to go near houses.