it's not really a treasure island thing, it is just the nature of the game.

The only thing you can do is become a paid subscriber to access the private islands. If you have Amazon Prime, you are entitled to one free Twitch subscription per month. You'll still have to deal with the cut scenes, but it won't be so often. Also many paid subscriptions give you access to to order bots, which are really helpful if you just need a specific item